See this other answer of mine for a full derivation of the general form of a unitary $2\times2$ matrix.
As shown there, unitary matrices (notice I'm not imposing a constraint on the determinant here) can be written as
$$U=\begin{pmatrix}e^{i\alpha_{11}}\cos\theta& e^{i\alpha_{12}}\sin\theta\\
e^{i\alpha_{21}}\sin\theta & e^{i\alpha_{22}}\cos\theta
\end{pmatrix},\tag A$$
with the parameters $\alpha_{ij}$ satisfying the condition
$$\alpha_{11}-\alpha_{12}=\alpha_{21}-\alpha_{22}+\pi.\tag B$$
There are then many different ways to choose how to write these coefficients. This freedom mostly arises from the fact that a global phase change of the matrix doesn't change anything in the physics, and thus $U\simeq e^{i\phi}U$ for any $\phi\in\mathbb R$.
One relatively standard way to fix a notation is to fix the determinant. In the above notation, this determinant reads
$\det U=e^{i(\alpha_{11}+\alpha_{22})}$,
where I've used (B) and standard trigonometry to simplify the coefficients.
Imposing $\det U=1$ thus corresponds to the constraint $\alpha_{11}+\alpha_{22}=0$ (or $2k\pi$ for $k$ integer of course, but nothing changes if we pick $k>0$ here). Using this again in (B) then also implies $\alpha_{21}+\alpha_{12}=-\pi$.
With this added constraint, the $U$ now reads
$$U=\begin{pmatrix}e^{i\alpha_{11}}\cos\theta& e^{i\alpha_{12}}\sin\theta\\
-e^{-i\alpha_{12}}\sin\theta & e^{-i\alpha_{11}}\cos\theta
\end{pmatrix}.$$
Equivalently, I can say that a general SU(2) matrix has the form
$$U=\begin{pmatrix}x& y\\ -\bar y & \bar x\end{pmatrix},\tag C$$
with $x,y\in\mathbb C$ satisfying $|x|^2+|y|^2=1$.
Now, if I want to describe a general unitary matrix, and not just one with unit determinant, I can simply write $U=e^{i\phi}(e^{-i\phi}U)$ where $\det U=e^{2i\phi}$ (remember that the determinant of a unitary matrix is always a phase). Then, $\tilde U\equiv e^{-i\phi}U$ has unit determinant (remember that $\det(\lambda U)=\lambda^2 \det U$), and can thus be written as per (C).
Putting the above results together, we conclude that a generic unitary matrix has the form
$$e^{i\varphi}\begin{pmatrix}x & y\\ -\bar y& \bar x\end{pmatrix},\tag D$$
with the additional constraint $|x|^2+|y|^2=1$, and arbitrary phase $\varphi\in\mathbb R$.
What is nice about (D) is that it gives us a pretty straightforward recipe to figure out how the minuses and other phases should be placed: simply put out phases from the matrix in order to make it into a unit-determinant one, and then verify that what is left looks like (C).
Applying this to your example, we get
$$\begin{pmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin(\frac{\theta}{2})\\e^{i\phi}\sin(\frac{\theta}{2})&e^{i\phi+i\lambda}\cos(\frac{\theta}{2})\end{pmatrix}=
e^{i(\phi+\lambda)/2}
\begin{pmatrix}e^{-i(\phi+\lambda)/2}\cos(\frac{\theta}{2})&-e^{i(-\phi+\lambda)/2}\sin(\frac{\theta}{2})\\e^{i(\phi-\lambda)/2}\sin(\frac{\theta}{2})&e^{i(\phi+\lambda)/2}\cos(\frac{\theta}{2})\end{pmatrix},$$
which therefore looks like it should.
It is also worth noting here that this is not a general parametrisation of a $2\times 2$ unitary matrix. Global phase aside, a general parametrisation involves three parameters, not two. It is, however, a general parametrisation of unit determinant unitaries, identifying $U\sim e^{i\alpha}U$.
