I think a large part of your confusion can be resolved by understanding how to think about $\sigma$-algebras as "containing information". Consider the probability space $(\Omega,\mathcal F,\mathbb P)$, and the random variable $X$ on this probability space. Intuitively, you should imagine $\omega \in \Omega$ being drawn according to the measure $\mathbb P$, and the realised $\omega$, in turn, determines $X$.
When we say that we "know" the information contained in $\mathcal F$, you should think of this as being able to take any set $E \in \mathcal F$, and being able to determine whether $\omega \in E$ or $\omega \notin E$. (Now is a useful time to recall the definition of a $\sigma$-algebra.)
Since $X$ is a random variable, it must be $\mathcal F$-measurable. Intuitively, what this means is that the information contained in $\mathcal F$ must fully determine the random variable $X$. Once we know the content of $\mathcal F$, we know exactly what the value of $X$ is. (Of course, the randomness of $X$ comes from the fact that we typically do not know $\mathcal F$, but only some sub-$\sigma$-algebra.) This captures the idea that the $\sigma$-algebra of the underlying probability space captures everything there is to know in the environment we are trying to model.
The conditional expectation $\mathbb E [X \vert \mathcal G]$ is our best guess of $X$, given that we know $\mathcal G$. Hence, $\mathbb E [X \vert \mathcal G]$ must be $\mathcal G$-measurable: the information contained in $\mathcal G$ must be enough to determine $\mathbb E [X \vert \mathcal G]$. This explains the first requirement of the definition. (Incidentally, regarding your second edit, your proposed conditional expectation works only when $Y$ is $\mathcal G$-measurable. If it isn't, then $Y$ is not (a version of) the conditional expectation with respect to $\mathcal G$.)
The second requirement captures the definition of the conditional expectation with respect to events. However, instead of choosing some specific event $G\in\mathcal G$, we allow the conditional expectation to vary over all possible events in $\mathcal G$. (Observe that if $\mathcal G$ were the $\sigma$-algebra generated by the event $G$, then this calculation is pretty much equivalent to calculating the conditional expectation with respect to an event.)
One advantage of conditioning with respect to $\sigma$-algebras rather than events is that it gives you a flexible apparatus for reasoning about things you might know in the future before you actually discover this knowledge. For example, suppose that I will know $\mathcal G$ tomorrow, then I can write down what my best guess of some random variable at that time will be, without necessarily waiting to see which event in particular is realised. (I imagine someone better versed in stochastic analysis than I am will be able to give you a better reason/example, but hopefully this is at least somewhat helpful.)