Where are these definitions from? I was unable to understand what you mean by a "reconstrual extension".

Maybe you're following section $4$ of The $2$-categorical structure of predicate theories? That's the best source guess I can make given the terminology used in the OP.

@NoahSchweber Hi, thank you for that, I added the relevant excerpts from the book. Feel free to have a look.

@AlexKruckman Hi, I added the relevant excerpts feel free to have a look.

Sorry, I try to put it in my own words but it seems like it doesn't work.

@BertrandWittgenstein'sGhost What is the book? Is it The logic in philosophy of science?

@NoahSchweber That citation you gave seems very very similar, I believe they are the same thing except that the book I am referring to is for propositional logic and the other for predicate.

@NoahSchweber Yes. That's the book!

@BertrandWittgenstein'sGhost I still don't know what you mean by "an identity function," but I think you just want to talk about $1_T$ and $1_{T'}$.

The definition of "homotopy equivalent" is cut off in your image. Some rules of thumb: if you want to quote a definition, (1) quote it in the exact words, (2) cite the source clearly in the question, (3) take the time to type it out rather than embedding an image.

Note that in Chapter 4 the author re-develops everything for first-order logic anyways. On that note, the entire book is available essentially chapter-by-chapter at the philosophy of physics bootcamp site (see the "2020 summer, part 1" bit).

@AlexKruckman's suggestion of typing the definitions out yourself is good for the additional reason that that forces you to carefully read through them.

@NoahSchweber Thanks for that! You are correct, I shouldn't have called it an identity function. That said, I have fixed the question. If not, I will try retyping all the relevant definitions in here but that might take some time.

@AlexKruckman I am sorry, I realized that right now. It should be fixed. If not I will retype all the relevant definitions from that chapter in here, but that will take some time.

@BertrandWittgenstein'sGhost FYI the book ... isn't great about this. The only discussion of the use of the term I can find is on page $118$ (for the first-order version, which the author has switched to at the beginning of chapter 4), which doesn't have much content at all. I suspect that the motivation for the term is not topological but rather categorical - it behaves the way the usual notion homotopy equivalence does in the context of categories of "nice" spaces or similar objects. (See e.g. here for some categorical generalizations.)

I am not either able to make sense of the given définitions, also because I am not a logician at all. However, the notion of being homotopy équivalent is much more general than the situation in spaces! Think for example of the notion of two chain complexes being equivalent. In general what you need is a "cylinder object" $Cyl(X) $ with two equivalences $i_0, i_1: X \to Cyl(X) $. In the case of spaces $Cyl(X) = X \times I $ and in the case of complexes is $X \otimes J$ for some "interval object" $J$.

All this stuff takes place in what is called "model category theory", but I personally doubt that the framework you cited has so much structure. I guess it is rather an intuitive similarity, and finding a " trivial deformation" $Cyl(X)$ with two copies of $X$ inside , such that $i_0, i_1 : X \to Cyl(X) $ are homotopy equivalences in the sense you cited is a starting point. Two maps are then said to be equivalent if $f_0, f_1:X \to Y$ factors through $h: Cyl(X) \to Y$ with $f_0=hi_0, f_1=hi_1$. Homotopy equivalence of objects $X \simeq Y$ is then $f:X \to Y, g:Y \to X$ st $fg ~ 1, gf ~1$.

@NoahSchweber Thank you for that, I will try wrapping my head around it and see if I can form an intuition of sorts.

@AndreaMarino Thanks for the information. I am not familiar with much of what you said, but it's interesting. Can you refer me to some text that can further elaborate on that?

@NoahSchweber The author of this text defined 'model' as follows: "For a set delta of sigma sentences, we say that v is a model of delta just in case v(φ)=1, for all φ in delta." That person also defined 'signature' as a collection of items that they call propositional constants.