5

I don't think this terminological issue is as recent as you ask for, or arises in exactly the way you describe, but let me give the example anyway.
The index of an algebraic variety $X$ with canonical (Weil) divisor $K_X$ is the smallest natural number $n$ such that $nK_X$ is a Cartier divisor. An example of this usage is in this paper of Fujino.
But also:...

2

In the same vein as my response to your other question, if pointed finite sets are an eleutheric system of arities, Lawvere theories over that system of arities will be equivalent to monads in a certain monoidal category. This is in section 11 of Lucyshyn-Wright here.
Edit: I briefly outlined how the relationship between eleutheric systems of arities and ...

2

Rather than checking to see if that is a substitution monoid, I think you might have an easier time using Rory Lucyshyn-Wright’s notion of a eleutheric system of arities, seen here. It’s a relatively straightforward condition to check, with several equivalent statements.
Edit: To give a (very brief) description of how this works out: a monoidal subcategory ...

2

It is worth mentioning that category theory can be articulated through and founded on sets, in a relatively straightforward manner.
That said, category theory and set theory seem to be two sides of the same coin even at a research level. I asked about this comparison here and got some excellent discussion from category and set theorists (see the comments).
...

2

One example that I've seen is the use of the word "synthetic," which has multiple uses in differential geometry.
There is a field called synthetic differential geometry, which studies differential geometry from the viewpoint of topos theory. This is based off work of Lawvere, and popular among the more categorically minded; the ncat lab describes it here.
...

1

The word “topological stack” has at least three usages:
A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack if there is a a morphism of stacks $p: \underline{M}\rightarrow \mathcal{D}$ for some manifold $M$, such that $p$ is a representable epimorphism. This is Definition 2.22, page number 86 in David Carchedi’s thesis.
A stack $\...

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