I suspect that the question is grounded in the false visualization of superpositions as mixtures, so I think we need to do a deep dive on the question and the issue underlying it to get to the root of the issue. It would be more accurate to say that a mixed state represents being "two places at once", than a superposition does, and I'll have more to say on that, at the end of the reply.
First: superpositions are not mixtures! Everything is "in a superposition", even things that are "not". For instance,
$$|0⟩ = \frac{\frac{|0⟩ + |1⟩}{\sqrt{2}} + \frac{|0⟩ - |1⟩}{\sqrt{2}}}{\sqrt{2}},\quad |1⟩ = \frac{\frac{|0⟩ + |1⟩}{\sqrt{2}} - \frac{|0⟩ - |1⟩}{\sqrt{2}}}{\sqrt{2}}.$$
To make this clear, think of $0$ as red and $1$ as blue. Then you have two superpositions of two forms of purple, in which one yields red and the other yields blue. You can do that, in optics, though you'll need lasers or other coherent light sources to do it with.
The question behind the question is: what, if anything, is happening when there is "wave-function collapse", which I will address further below. But, first things first.
School-children are taught wrong about the colors, when referring to light-colors (as opposed to paint-colors). Light colors are not mixtures - not even when they are composed of combinations of frequencies. Such states are just as pure as the frequency eigenstates are, and the "superposition-component" relation is not a part-whole relation, but is symmetric! In particular, to say "A is a superposition of B and C" is also to say that "B is a superposition of C and A" and "C is a superposition of A and B".
This wrong teaching leads to wrong visualization, even sub-consciously, by children when they grow up to become adults, even by those children who grow up to become physicists; including expert physicists. They may "know" in their heads that superpositions are not mixtures, but the real they, which is in the core of their being and not just in their heads, still carry on visualizing superpositions as mixtures and framing everything in terms of this wrong visualization - as you apparently just did.
In contrast, this is a mixture:
$$\frac{|0⟩⟨0| + |1⟩⟨1|}2.$$
As to what the essence of a "collapse" is: that's a different question entirely. The textbook form of the Copenhagen interpretation, as best as any such interpretation has been articulated that everyone agrees is a "Copenhagen interpretation", says that quantum systems go along happily on their merry way, with states that evolve according to a form of the Schrödinger Equation, when formulated in the Schrödinger Picture, or with kinematic and dynamic variables that unfold according to a form of the Heisenberg Equation, when formulated in the Heisenberg Picture. I use the term "unfold", because the Heisenberg Equation - when written out for use-cases in field theory - takes the form of partial differential equations that mostly match what the corresponding field equations look like in the limit of classical physics, with all space-time coordinates on an equal footing, rather than with the time coordinate singled out and treated specially.
In the Schrödinger Picture:
When a measurement takes place, according to the Copenhagen Interpretation, with the measurement being of attributes of the system undergoing measurement, that are represented by "operators", the result is an eigenvalue of the operator, and the resulting state is the corresponding eigenstate, with the probability determined by the coefficients of the decomposition of the system's initial state into a sum of the eigenstates. The transition is stochastic.
If the measurement were of colors, with the state initially being represented by
$$\frac{|0⟩ + |1⟩}{\sqrt{2}}$$
with $0$ being red and $1$ being blue, then the initial state, itself, which would be the following pure state
$$\frac{|0⟩ + |1⟩}{\sqrt{2}}\frac{⟨0| + ⟨1|}{\sqrt{2}} = \frac{|0⟩⟨0| + |1⟩⟨1|}2 + \frac{|0⟩⟨1| + |1⟩⟨0|}2,$$
would lead to one of the two pure eigenstates
$$|0⟩⟨0|,\quad|1⟩⟨1|$$
each with probability $1/2$. That's actually the essence of what the mixed state
$$\frac{|0⟩⟨0| + |1⟩⟨1|}2$$
is! So, the transition enacted by the measurement could be represented by
$$\frac{|0⟩⟨0| + |1⟩⟨1|}2 + \frac{|0⟩⟨1| + |1⟩⟨0|}2 → \frac{|0⟩⟨0| + |1⟩⟨1|}2,$$
possibly with an additional, stochastic, stage that forks off as
$$
|0⟩⟨0| \overset{½}{←} \frac{|0⟩⟨0| + |1⟩⟨1|}2 \overset{½}{→} |1⟩⟨1|
$$
with the respective probabilities labeled, depending on whether or not you interpret the mixed state as already connoting that branching.
Either way, the process is an application of the Born Rule. For a sequence or web of measurements, this generalizes to Lüder's Rule, both rules being representations of the Projection Postulate.
The actual dynamics of the transition, in realistic applications, is captured by the Lindblad Equation, which describes open quantum systems that are subsystems of quantum systems (e.g. a quantum system undergoing measurement within a larger environment where the total system is a quantum system) and ... if you believe Oppenheim's new Post-Quantum Gravity paradigm ... also at a fundamental level for hybrid classico-quantum systems.
In Oppenheim's formulation, the very existence of stochasticm arises as a self-consistency requirement for hybridizing classical and quantum systems. Once you have the existence of stochasticism established, then the Born Rule and its generalization, Lüder's Rule, follow by Gleason's Theorem.
You will have noticed that I conditioned all of this on the Schrödinger Picture, but haven't said what the Projection Postulate looks like in the Heisenberg Picture. That's because there's a gap there! There really isn't a consensus, or standard text-book treatment account of projection in the Heisenberg Picture, because in that formulation of quantum theory, the states are timeless, while it is the operators, that represent the system's attributes, that undergo change and unfolding. Something like the Lüder's Rule would be involved in a formulation of projection in the Heisenberg Picture, because all the measurements are treated in tandem by the rule, rather than sequentially in time in a linear or partial order. In contrast to the "moving time universe" view of time embodied within the Schrödinger Picture, the Heisenberg Picture is the "block universe" view of time. Some references on the "Block Universe" concept: The Block Universe: Understanding Time as a Dimension, The Block Universe Theory: Is Time an Illusion or a Multiversal Playground?, A Critical Examination of the Block Universe Theory. Part of the obstruction to a formulation of the Projection Postulate in the Heisenberg Picture is that Lindbladian dynamics doesn't really have a sensible Heisenberg Picture formulation, as alluded to in Appendix B of the above-mentioned Oppenheim link.
It is possible to repair that deficiency, with a compromise "Lindblad Picture", that would better embody the idea of a Growing Block Universe, or more accurately: a sideways-shifting / forking block universe, in which the effective Heisenberg Picture state undergoes a kind of side-ways transition under the effect of the non-unitary part of the Lindblad equation. It serves as a hybrid, compromise picture analogous to the Interaction Picture of quantum field theory.
This reply is really a follow-up to my earlier reply to Are standard QFT and general relativity contradictory?, and I also put up a biblio and archive Classico-Quantum that includes refurbished LaTeX sources for some of the open access versions of the published links, including the one above for Oppenheim, as well as others. If I have enough time, I may add an Appendix C to my copy it that describes the formulation and interpretation of the Lindblad Picture.
As a preview, here: (The Lindblad Picture, fleshed out), using the Chat GPT as a sounding board to flesh out the math and ideas behind the Lindblad Picture, you can see how it fixes a lot of interepretational problems with quantum theory, especially that of devising a measurement theory for the Heisenberg Picture that fixes the problems that Deutsch's attempt at this had in it.
Now, back to your query:
If you think of the 0 and 1 states as being locations, instead of colors, then to more directly address the main question behind your query, you might ask: what does a superposition of position eignstates, such as $\sqrt{½} (|0⟩ + |1⟩)$ look like and where is it located? If I said it is located nowhere at all, because it's not a position eigenstate, you might counter that we could measure it as follows.
We could prepare a large number of objects in this state, measure each one, the position measurement being like a snapshot, and build up a multiple-exposure picture of the state. Half of the time, following a position measurement, $\sqrt{½} (|0⟩ + |1⟩)$ will show up as $|0⟩$, while the other half of the time it will show up as $|1⟩$. The end result is that you will get a multiple exposure picture that's evenly split between positions 0 and 1.
But that's exactly what the mixed state
$$\frac{|0⟩⟨0| + |1⟩⟨1|}2$$
is and describes! In contrast, the superposition corresponds to this state:
$$\frac{|0⟩⟨0| + |1⟩⟨1|}2 + \frac{|0⟩⟨1| + |1⟩⟨0|}2.$$
Its "location" wasn't measured at all. It just doesn't have any location, because it's just not a position eigenstate. The measurement turned that state into the mixed state and that's what showed up in your multiple-exposure snapshot. Confusing the mixed state seen in the snapshot for the superposition it came from is where the cardinal sin of confusing mixed states and superpositions happens.
If you think of snapshot being a detection screen, then each of the Born Rule transitions
$$
\frac{|0⟩⟨0| + |1⟩⟨1|}2 + \frac{|0⟩⟨1| + |1⟩⟨0|}2 → \frac{|0⟩⟨0| + |1⟩⟨1|}2 \overset{½}{→} |0⟩⟨0|,\\
\frac{|0⟩⟨0| + |1⟩⟨1|}2 + \frac{|0⟩⟨1| + |1⟩⟨0|}2 → \frac{|0⟩⟨0| + |1⟩⟨1|}2 \overset{½}{→} |1⟩⟨1|
$$
will produce a dot on the detector screen; one at location 0, if the final outcome is $|0⟩⟨0|$, and one at location 1 if the final outcome is $|1⟩⟨1|$. Slowly, you're building up a picture of the mixed state that these two outcomes branched from
$$
\frac{|0⟩⟨0| + |1⟩⟨1|}2
$$
and in this way you may endow the mixed state with an operational meaning.
The best you can say is that, for a given operator $x$, there is an equivalence relation between states, such that if two states $ρ_0$ and $ρ_1$ are equivalent $ρ_0 \overset{x}{≡} ρ_1$ if their respective matrix expansions $ρ_{0x}$ and $ρ_{1x}$ in the eigenbasis of $x$ have the same diagonal components. Then, you could say that
$$
ρ_0 = \frac{|0⟩⟨0| + |1⟩⟨1|}2 + \frac{|0⟩⟨1| + |1⟩⟨0|}2 \overset{x}{≡} \frac{|0⟩⟨0| + |1⟩⟨1|}2 = ρ_1
$$
where $x = |0⟩ x_0 ⟨0| + |1⟩ x_1 ⟨1|$ is the position operator with position eigenvalues $x_0$ and $x_1$. Only then you could say that the superposition state $ρ_0$ "has the potential of being" $½$ at $x_0$ and $½$ at $x_1$, matching its $x$-equivalent position eigen-state $ρ_1$. The relation $\overset{x}{≡}$ could then be called a "potential of being" relation for operator $x$. Nonetheless, the state $ρ_0$ has no position, per se. Only $ρ_1$ does.
