Can pure qubit states be represented in a unit circle in $\mathbb R^2$? How does such representation relate with the Bloch sphere? In this video they are regarding 2D circles denoting real-valued states of qubit, like

Teacher says it can be extended to 3D to Bloch sphere.

But Bloch sphere has |0> at the top and |1> at the bottom, while 2D circle has |0> also at the top, while  it has |1> at the right. How these can be corresponded?
 A: I think what they are doing is representing the (real part of) the complex vector space in which a (pure) qubit is embedded.
Remember that the state of a $d$-dimensional system, which we usually denote via a complex unit vector $|\psi\rangle\in\mathbb C^n$, is more precisely defined as an equivalence class of such vectors, where different unit vectors are identified when they differ by a global phase.
Formally, we call the corresponding set of equivalence classes a complex projective space $\mathbb{CP}^{n-1}$.
The Bloch representation (the Bloch sphere, for a single qubit) is a way to represent such complex projective space as a subset of $\mathbb R^{n^2-1}$.
It is worth stressing that this means that there is a one-to-one relation between states and representative points in the Bloch representation.
What they seem to be using in the video is a bit different. Rather then using the Bloch representation, they are directly representing (pure) states via the corresponding point in the embedding vector space. In other words, they represent $|\psi\rangle=\sum_i \psi_i |i\rangle$ as the vector $(\psi_1,...,\psi_n)\in\mathbb C^n$.
Restricting to $n=2$ (a single qubit) and considering only states with real coefficients, we thus have the set of states of the form
$$a |0\rangle + \sqrt{1-a^2} |1\rangle \simeq \begin{pmatrix}a\\ \sqrt{1-a^2}\end{pmatrix}\in\mathbb R^2, \qquad a\in[-1,1].$$
These are the vectors shown in the screenshot. Not that in this representation, $|0\rangle\to (1,0)$ and $|1\rangle\to (0,1)$, compatibly with the screenshot.
However, this representation has the severe shortcoming that different vectors correspond to the same state. For example, you might notice how $(0,1)$ and $(0,-1)$, which both represent the state $|1\rangle$, are assigned different points in the picture.
You can imagine every point in the Bloch representation as a direction in their "unit circle state machine" representation.
