Understanding of extension fields with Kronecker's thorem

Question 1) Yes, the definition in Gallian is equivalent to "$F$ is a subfield of $E$." When he says that the operations of $F$ are the operations of $E$ restricted to $F$, that does indeed mean that you can add/subtract/multiply elements of $F$ just by viewing them as elements of $E$ and performing the same operations.

Question 2) I like that you are being picky here, and making sure that in the definition is indeed satisfied. Although Gallian does not explicitly say so here, you can identify $F$ with the set of cosets of constant polynomials, and now you really do have a copy of $F$ as a subfield of $E$.

Your alternative definition looks pretty good, but you need specify that $\phi$ is a nonzero homomorphism (unless you require homomorphisms to send 1 to 1, in which case this is already guaranteed). Also, since you have a homomorphism, the operations on $\phi(F)$ are automatically those inhereted from $E$.

The main thing is that every homomorphism between fields is an embedding (injective). On one hand it means that your definition is correct (assuming that homomorphisms take $1\mapsto 1$, as Brett said), and yes, more precise. On the other hand, it means that if we have a homomorphism $K\to L$, then, basically we can assume that $K\subseteq L$ is a subfield (in this step we identify $K$ with its image), and this viewpoint has some advantages, at least in simplifying the notations.