On the one hand, you sort of don't need to worry about it, but it's best to make your advisor happy (especially if they are a co-author). Here are my thoughts.
Your advisor ought to have experience with this sort of thing in general. So as a rule, it's good to take their opinion into consideration.
The journal you ultimately submit to will have people whose job is to make sure you use any nuanced writing rules like this correctly. And the formatting and such that you choose will probably be changed by the journal any way.
I've never heard of any rule like you mention, and I don't think there's any expectation in the community about this. Citations in math papers are always weird. They have no standard format (the journal will do the format they like for you), and you don't have to worry so much about exactly quoting somebody per se.
My rule of thumb is that when I cite something, I always try to credit previous literature for everything I can. I ask myself "is what I'm writing obviously equivalent to or implied by the theorem they wrote?" If so, I would simply phrase the result in my own words and credit them for the theorem. You can do this easily, and you don't have to mention a proof at all unless it's unclear how their result implies your statement.
For example, I would write something like:
We recall the following theorem due to Smith [4].
Theorem 3: All objects have property.
And the theorem number is just whatever number would be next in your paper. Or you could do
The problem was originally asked by Smith [4], who addressed the situation when things are nice. In particular, she proved.
Theorem 3 (Smith [4]): All objects have property.
Or if it's very hard to find where in [4] Smith wrote that theorem, you could say
Smith proved the following, which appears as theorem 13.b of [4]
Theorem 3: All objects have property.
Or if you want to emphasize that what you're stating is easily implied by Smith, I might say something like
We now use a special case of a theorem of Smith [4] recalling here only what we need.
Theorem 3: The objects I like have property.
Note that in all of the above cases, it wouldn't be appropriate to cite a proof or anything after the theorem statement. The result should either be very obviously the same as the paper you cite, or it should be very obviously implied by it. If you want to use something similar to Smith's theorem, but it's not obviously the same or implied, then I suggest something like the following.
We now appeal to a general theorem of Smith [4], which states
Theorem 3: All objects have property.
By regarding her (thing) in (our setting), we obtain (by this hint of a proof sketch)
Corollary 4: The things we care about do the thing we care about.
If needed, then say more about why this follows from Smith's result. If this connection (and stating Smith's result in the first place) is too much, then you might want to consider putting that part in an appendix and just saying something like "the following is obtained from a theorem of Smith [4], but for ease of reading this connection is discussed in appendix A."
One final piece of advice would be to simply read a lot of well-written papers and see what they do. Look in places that have good exposition even if it's not in your area. It's not important to understand the math in those articles so much as how they write and how they handle these nuances.
You have a lot of freedom, and any choice you make is likely going to be fine as long as you thoroughly credit previous authors and you make your paper easy to read.
