I don't want to quibble about names (and I wasn't the person to come up with the term "baby logic"), but the point is that just introducing a couple of connectives and proof strategies doesn't even constitute the basics of what mathematical logic really is about. Which btw I'm perfectly fine with, most people don't need more than that.

If you do want to study logic formally, the basics start with well-formed formulas, signatures, etc.

I guess what you call it doesn't matter a lot, but the discussion seems to have started with the assertion that most students, even in mathematics, never really learn formal logic, and I would agree with that (under my definition of "formal logic"), while also agreeing with you that you can't pursue a degree in maths without knowing how induction works or what a bijection is. But still, most people don't need to know exactly how to formalise induction and that it's actually (in its full form) a second-order axiom.