Can a Formal Language be of a type (RE, REC, Regular, etc) for one TM, but of a different type for another?
A language $\mathcal L$ is recursive if there exists a Turing machine $\mathcal M$ (and therefore, an algorithm) that stops on every input, and that accepts exactly words from $\mathcal L$ (i.e. $\forall w, \mathcal M$ accepts $w\iff w\in\mathcal L$). Designing a Turing Machine that doesn't comply with those conditions won't help you prove that there isn't another Turing Machine that actually works as needed. Therefore, the recursivity of a language doesn't depend on a "choice" of a Turing Machine.

However, to prove a language is recursive, you usually find a Turing Machine that falls under our definition. We denote such a Turing Machine as a witness, or you could say that it witnesses that your language is recursive* 



Moreover, classes of languages are often included in each others. By instance, if a language $\mathcal L$ is regular, then there is a DFA that recognizes it, and therefore witnesses the regularity of $\mathcal L$. But if there is such a DFA, you can easily build a Turing machine that "simulates" that DFA, stopping of every input, and accepting exactly the same words as the DFA. Therefore, $\mathcal L$ is REC as well, as you pointed out. But this shouldn't lead you to make more assumptions than needed. When you read "$\mathcal L$ is recursive", you shouldn't make the assumption that it isn't regular but you shouldn't make the assumption that it is either. You simply don't know.



As all of this suggests, it is usually easier to prove that a language belongs to a class, than to prove it doesn't. To prove belonging, you only have to provide a witness. To prove that $\mathcal L$ doesn't belong, you basically have to prove that no TM/DFA/... can be a witness. This is usually relies on "tricks", as the diagonal argument (you should look into the Halting problem for more information) for recursivity, or the pumping lemma for regularity.

(*) : Sometimes, it will be more convenient to use the fact that $Rec = $R.E.$\cap$ co-R.E., but this relies on a constructive proof, which means this is a "mechanical" way to build a witness for recursivity when you have both a witness for recursive enumerability and corecursive enumerability