I am familiarizing myself with the notion of a tensor product. I believe I have a clear idea on what the tensor product itself is. That is, for vector spaces $V_1, \ldots, V_k$ $$V_1 \otimes \cdots \otimes V_k = \Big\{\sum_1^n a_i (v_1, \ldots, v_k) : v_i \in V_i, a_i \in \mathbb{F}, n \in \mathbb{N}\Big\}\Big/Q$$ where $\mathbb{F}$ is some scalar field and $Q$ is a set we quotient out to obtain the desired properties of a product (scalar multiplication, linearity, distributivity).
If I recall correctly, I have also seen tensor products defined as $$V_1 \otimes \cdots \otimes V_k = \Big\{\sum_1^n a_i(v_1 \otimes \cdots \otimes v_k) : v_i \in V_i, a_i \in \mathbb{F}, n \in \mathbb{N}\Big\}$$ where the operation $v_1 \otimes \cdots \otimes v_k$ is defined to satisfy the desired multiplication properties we previously quotiented out.
What I am confused about is what exactly is the operation $v_1 \otimes \cdots \otimes v_k$ (I believe these are called pure tensors), defined on individual vectors as opposed to spaces? It seems one is used to define the other, and so the circular definition is causing me some trouble. Is $$v_1 \otimes \cdots \otimes v_k = (v_1, \ldots, v_k)\big/Q$$ where $v_i \in V_i$? However it wouldn't make sense to quotient out a single element, so what exactly do we mean by the image of $(v_1, \ldots, v_k)$ (which is said to equal $v_1 \otimes \cdots \otimes v_k)$ under this quotient?
Another definition is from Lee's text on smooth manifolds, where he states $$V_1 \otimes \cdots \otimes V_k = F(V_1 \times \cdots \times V_k)\Big/Q$$ where $F$ is the free vector space on $V_1 \times \cdots \times V_k$ (defined as the set of all formal linear combinations of elements of $V_1 \times \cdots \times V_k$). He defines $$\Pi: F(V_1 \times \cdots \times V_k) \rightarrow V_1 \otimes \cdots \otimes V_k$$ to be the so called "natural projection". Using this idea he says the equivalence class of an element $(v_1, \ldots, v_k)$ in $V_1 \otimes \cdots \otimes V_k$ is denoted by $$v_1 \otimes \cdots \otimes v_k = \Pi(v_1, \ldots, v_k).$$ What exactly is the natural projection here, I have not seen this terminology before.
As an example, let us take $v_1 = (1,0) \in \mathbb{R}^2$ and $v_2 = (1, 0, 0) \in \mathbb{R}^3$. How would one explicitly calculate this tensor product?