Recommendation on a rigorous and deep introductory logic textbook

There are two very different kinds of question here:

What is logic exactly? ... What is "proof"? What is "truth"?

All good questions. But famously they do not have sharp, determinate, clear, uncontentious answers. Indeed, they are characteristically philosophical questions (that fall into the purview of what is often called "philosophical logic").

Of course, a technical logic text will introduce e.g. a sharp, technical, notion of a proof-in-a-given-formal-system (the fine print can be significantly different in different texts). But what is the relation between (1) the everyday notion of mathematical proof and (2) various notions of proof-in-a-given-formal-system which aim to model mathematical proof? This is up for (philosophical) debate. Similarly for the notion of truth, and indeed for the notion of a logic.

A "rigorous logic text" is therefore not the best place, really, to look for the discussion of the philosophical questions here. For those questions are (as it were) standing back from details in those rigorous texts and asking more general, philosophical, questions about them.

Please recommend me a good precise logic textbook.

Still, if you do want pointers to formal logic textbooks then there are a lot of suggestions, at various levels, on various areas of logic, in the Guide you can find at http://www.logicmatters.net/tyl