The paraphrase is attempting to be more rigorous than the given definition, but it is not rigorous enough, so you are not really resolving your confusion.
Let's start with something a bit simpler: suppose you have sets $A$ and $B$, and a function $f\colon A\to B$. Given any subset $A'\subseteq A$, you can define the restriction of $f$ to $A'$, often denoted $f|_{A'}$. This is a function $f|_{A'}\colon A'\to B$ whose definition if $f|_{A'}(a') = f(a')$ for every $a'\in A'$. That is, the restriction of $f$ differs from $f$ by virtue of having a different domain, but othwerwise takes the same values.
If $g\colon A\to B$ is a function, $B'$ is a subset of $B$, and $g(A)\subseteq B'$, we can also define the co-restriction of $g$ to $B'$ (I am not aware of any standard or common notation for this, however), as $g'\colon A\to B'$, where $g'(a)=g(a)$ for every $a\in A$. When you care about both domain and codomain of a function, this is a function that is different from $g$ because it has different codomain.
If $\mathbf{V}$ is a vector space over $F$, then vector addition is a binary operation on $\mathbf{V}$, and hence a function $+_{\mathbf{V}}\colon\mathbf{V}\times\mathbf{V}\to\mathbf{V}$. We usually use infix notation for operations, rather than the traditional (in the United States, at any rate) prefix notation for functions. So we write "$\mathbf{x}+_{\mathbf{V}}\mathbf{y}$" to mean the result of applying the function $+_{\mathbf{V}}$ to $(\mathbf{x},\mathbf{y})$. That is, instead of writing the result of using the function as $+_{\mathbf{V}}(\mathbf{x},\mathbf{y})$, we write $\mathbf{x}+_{\mathbf{V}}\mathbf{y}$. Similarly, scalar multiplication is a function $\cdot_{F,\mathbf{V}}\colon F\times\mathbf{V}\to\mathbf{V}$, and we usually denote $\cdot_{F,\mathbf{V}}(\alpha,\mathbf{x})$ by $\alpha\cdot_{F,\mathbf{V}}\mathbf{x}$, or just $\alpha\mathbf{x}$.
If $\mathbf{W}$ is a subset of $\mathbf{V}$, then $\mathbf{W}\times\mathbf{W}$ is a subset of $\mathbf{V}\times\mathbf{V}$, so we can consider the restriction of $+_{\mathbf{V}}$ to $\mathbf{W}\times\mathbf{W}$. And since $F\times\mathbf{W}$ is a subset of $F\times\mathbf{V}$, we can consider the restriction of $\cdot_{F,\mathbf{V}}$ to $F\times\mathbf{W}$.
Suppose that we do that, and we have that whenever $\alpha\in F$, $\mathbf{x},\mathbf{y}\in \mathbf{W}$, we also have $\alpha\cdot_{F,\mathbf{V}}\mathbf{x}\in\mathbf{W}$ and $\mathbf{x}+_{\mathbf{V}}\mathbf{y}\in\mathbf{W}$. Then we can also consider the co-restrictions of $+_{\mathbf{V}}$ and $\cdot_{F,\mathbf{V}}$ to $\mathbf{W}$. Let us write this as $\overline{+_{\mathbf{V}}|_{\mathbf{W}\times\mathbf{W}}}$ (the co-restriction of the restriction to $\mathbf{W}\times\mathbf{W}$, and $\overline{\cdot_{F,\mathbf{V}}|_{F\times\mathbf{W}}}$.
With me so far? Okay, then the definition of "subspace" says:
If $(\mathbf{V}. +_{\mathbf{V}}, \cdot_{F\mathbf{V}})$ is a vector space over $F$, and $\mathbf{W}\subseteq \mathbf{V}$ is a subset of $\mathbf{V}$, then we say that $\mathbf{W}$ is a subspace of $\mathbf{V}$ if and only if $(\mathbf{W},\overline{+_{\mathbf{V}}|_{\mathbf{W}\times\mathbf{W}}}, \overline{\cdot_{F,\mathbf{V}}|_{F\times\mathbf{W}}})$ is a vector space over $F$.
(And the clutter probably lets you see why we usually try to paraphrase this rather than make the notation be very accurate...)
About your thoughts:
No. Note that while the restriction of $+|_{\mathbf{C}^n}$ to $\mathbb{R}^n$ does work (meaning, you can also do the co-restriction), the same is not true of $\cdot|_{\mathbb{C},\mathbb{C}^n}$: if you try to restrict it from $\mathbb{C}\times\mathbb{C}^n$ to $\mathbb{C}\times\mathbb{R}^n$, you can no longer also co-restrict it: for instance, $i\cdot (1,1,\ldots,1)$ does not lie in $\mathbb{R}^n$. So this restriction is not a "scalar multiplication" on $\mathbb{R}^n$. However, if you consider $\mathbb{C}^n$ as a vector space *over $\mathbb{R}$, then you do get that $\mathbb{R}^n$ is a subspace of $\mathbb{C}^n$ over $\mathbb{R}$.
The problem with your proposal in item 2 is that you do not explain what $+|_{\mathbf{W}}$ and $\cdot|_{F,\mathbf{W}}$ mean. You would need to say something like "where by $+|_{\mathbf{W}}$ we mean blah", etc.
The idea of the definition is that $\mathbf{W}$ should be a vector space in its own right, when you add vectors in $\mathbf{W}$ exactly the same way as you add them when you consider them as elements of $\mathbf{V}$. This is analogous to how when we add two integers as if they were fractions we get the same answer as when we add them as integers. We usually think of addition of rationals as extending the addition of integers (we already know how to add integers, we are not using that to explain how to add rationals). But if for some reason we knew how to add rationals but not how to add integers, we could ask whether when you add two fractions-correspond-to-integers we get a fraction-that-corresponds-to-an-integer, and if so restrict the addition-of-rationals to integers to get a way to add integers.
Best Answer
Take $V=\Bbb R^2$, with its standard vector space structure over $\Bbb R$. Now, consider the bijection$$\begin{array}{rccc}f\colon&\Bbb R^2&\longrightarrow&\Bbb R^2\\&(x,y)&\mapsto&(x^3,y^3).\end{array}$$Consider on $\Bbb R^2$ the vector space structure such that the sum is $v\oplus w=f\left(f^{-1}(v)+f^{-1}(w)\right)$ and the product by a scalar is $\lambda\odot v=f\left(\lambda f^{-1}(v)\right)$. Then $\Bbb R\times\{0\}$ is a subspace of $\Bbb R^2$ with respect to both structures.