The state $|\mbox{+}i\rangle$ in the Bloch sphere picture sits on the y-axis ($\theta =$ 90° degree and $\phi =$ 90°) while $i|1\rangle$ is on the x-axis; and $i|1\rangle$ is just $|1\rangle$ with global phase $i$ ($\theta =$ 180° and $\phi =$ 90°).
However, both $i|1\rangle$ and $|\mbox{+}i\rangle$ look very similar to me. Based on this example, how do we interpret that something is a global phase?
It is just the convention that people use the notation $|1 \rangle $ to represent the vector $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ and $|0 \rangle$ to represents the matrix $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$. Similarly, people use the notation $|i\rangle $ to represent the vector $\dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix}$ .
I could have very much use a different notation, say $|k\rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix}$, instead. Maybe this would lessen the confusion.
So by looking at the vector itself, it would be less confusing. That is, if you look at the state $i|1\rangle$ you have
$$ i|1\rangle = i \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
which you can see that $i$ here is the global phase. But if you look at the state $|i\rangle$ you have
$$|i \rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix} = \dfrac{1}{\sqrt{2}} \bigg[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} + i \begin{pmatrix} 0 \\ 1 \end{pmatrix} \bigg] = \dfrac{1}{\sqrt{2}}\bigg[ |0 \rangle + i|1\rangle \bigg]$$
as you can see here, $i$ is not a global phase but rather a relative phase.
