Are partial recursive functions analogous to recursive languages or r.e. languages?
Let me quote from the second edition of the same textbook, section 9.1 (sorry, I don't have the first one):


  Recall that a language L is recursively enumerable (abbreviated RE) if L = L(M) for some TM M. Also, we shall in Section 9.2 introduce "recursive" or "decidable" languages that are not only recursively enumerable, but are accepted by a TM that always halts, regardless of whether or not it accepts.


Therefore recursive languages are associated to total recursive functions, not partial.

Hope this answers both the questions.

EDIT: to answer the comment:

On the one hand, the reason lies on intuition: the Church-Turing thesis is about capturing the intuitive idea of "effective computability" or "things that we can actually compute". Imagine a program that, if there is a correct answer then it computes the answer, otherwise it never halts (i.e. it gives an answer only if there is an answer). Would you say that such a program computes the answer or not? 

But on the other hand, if this intuitive picture doesn't convince you,  consider the following (way more concrete): consider an effective enumeration $(\varphi_i)_{i\in\mathbb{N}}$ of all total recursive functions and consider the function $\varphi:=x\mapsto\varphi_x(x)+1$. Now it is clear that you can compute $\varphi$: you can find the $x$-th function, you can compute it (as it is recursive) and you can sum 1 (as sum is recursive). But $\varphi$ cannot be total by a diagonal argument: if it were total than it would have its own index $i$, therefore

$$ \varphi(i) = \varphi_i(i)+1 = \varphi(i) +1 $$

which is a contradiction. This problem vanishes if you consider instead partial functions.