The use of the symbols $\implies$ and $\to$ can vary from book to book. Two different books can use a different symbol for the same concept, and two different books can use the same symbol for different concepts. So yeah, I can understand how this is all pretty confusing, because there is some ambiguity involved. The important thing is to always get clear on how a specific book uses a symbol.
In fact, even the expression 'formal proof' is ambiguous. In logic, this has a pretty clear definition: it is a demonstration that manipulates expressions in the language of formal logic, and where those manipulations have to follow very specific formal patterns. In mathematics, we are typically far less strict about the notation: you can use English phrases like 'Therefore', 'We now see that', etc. as part of the mathematical proof. Even individual statements can be phrased in English rather than some specific logical or mathematical language. Typically the thing that instructors/book require is merely that the proof follows a very clear organization and structure: something that very clearly lays out assumptions, and where you clearly indicate what follows from what: it is a 'formal' proof in that sense.
Indeed, to clearly indicate what follows from what, in math proofs we often use expressions like $1 \implies 2$ to indicate that 2 follows from 1 given the mathematical domain you are currently working with. This is closely aligned to the meta-logical usage of the symbol $\implies$, which denotes that some formal logic expression logically follows from some other formal logic expression. Indeed, we could try to formalize the domain with a bunch of axioms, and express those using a set $\Gamma$ of formal logic expressions, and if we also capture 1 and 2 using formal logic expressions $\phi$ and $\psi$ respectively, we can then make the purely meta-logical claim that $\Gamma, \phi \implies \psi$
Anyway, in both math and logic, the $\implies$ denotes a kind of reasoning that you do. It says: if you have [this], then you can (mathematically/logically) infer [that].
Also, in both math and logic, you can write things like $p \implies q \implies r$ as a shorthand way to say two things: $p \implies q$ and $q \implies r$.
When you were in your linear algebra class, you were definitely doing mathematical proofs, and I highly doubt you did formal proofs in the much stricter formal logic tradition. So for those proofs, the use of $\implies$ was probably completely appropriate. Also, it seems like the use of $\to$ in that particular context was apparently used to express some kind of mathematical operation on matrices. So that was indeed not at all used as a symbol of inference or reasoning, as you teacher indicated. Indeed, the $\to$ as used in the context of that linear algebra course is not at all like the $\to$ symbol you are finding in the logic textbook you are now working with: a good example of two different books using the same symbol for different concepts.
In the context of formal logic, the $\to$ is a truth-functional operator, also called the material implication. In this context, the expression $p \to q$ merely expresses a claim that would be considered false if $p$ is true, and $q$ is false, but true otherwise. This is a much weaker claim than saying that $q$ mathematically or logically follows from $p$.
Indeed, your $p \to q \to r$ is a case in point. As you point out, as a formal logic statement it is ambiguous: is it $(p \to q) \to r$ or $p \to (q \to r)$? And even if your text has a rule for parsing such statements a certain way (as Naim says in the Comments, some books will see $p \to q \to r$ as $p \to (q \to r)$, although many books do not have such a parsing convention and consider it simply to be an ungtrammatical expression), please note that neither $(p \to q) \to r$ nor $p \to (q \to r)$ is equivalent to $(p \to q) \land (q \to r)$. So no, this is completely unlike anything like the $\implies$.
OK, so what are you to use now with the logic textbook you are currently using? Well, as you found, your logic textbook depicts a proof as a long column of logic statements, the inferences are indicated by the use of a line number system: if line 6 refers to lines 3 and 5, then that is the formal proofs way of indicating that line 6 logically follows from lines 3 and 5. We could denote this as $3,5 \implies6$, but there is no further need for that. Indeed, from the quote in the second paragraph in your question, it looks like your logic textbook isn't going to make meta-logical claims using meta-logical symbols like $\implies$, which means that indeed, you should not be using that symbol, and just stick to $\to$ in the context of that book.
