The overall question: What are some good examples where a different understanding of terminology or notation caused you to misinterpret a result in a way that was inaccurate? The intent here is of course to provide some amusing anecdotes, but also to help your fellow mathematicians to avoid some of the pitfalls of historical documentation that previously tripped you.

Onto my first story. The problem arose while I was trying to gain a better understanding of the automorphism groups for the symmetric and alternating groups as outlined in W.R. Scott's Group Theory. A result in the book talked about an isomorphism from a group $G$ to a group $H$, but when I performed some cardinality computations, I discovered the order of $G$ was less than that of $H$. There was no way these two groups were isomorphic, they didn't even have the same size! I probably spent some 20 hours here and there over the next few months trying to reconcile this disparity. It turns out, the book uses the phrase "isomorphism into" (a group $H$) to mean what we in modern parlance call a monomorphism, and uses the phrase "isomorphism onto" to mean what we nowadays think of as an isomorphism.

The above story is indicative of terminology changing over time within a discipline, so that "text from then" and "text from now" had different meanings, but I have a second story that suggests history can cause a terminology issue in a different way: "text from research area A" versus "text from research area B" where the two research areas share common terminology stemming from a shared history, but with different meanings. The culprits in this story are group theory versus number theory.

Picture dozens of mathematicians together in a room to consult on a major cross-discipline mathematical project. At some juncture during the discussion, and after several minutes had passed since a prime factorization was given regarding the order of the algebraic object under discussion, I asked the presenter if the integer just computed was the square-free part of the order. The group theorists all agreed that it was, while the number theorists all agreed that it wasn't, and the next 30 minutes were spent trying to figure out the disparity.

It turns out that, while everyone agrees on what it means for a positive integer to be square-free, group theorists and number theorists disagree on what the square-free part of a given integer is. My understanding at that time was that the square-free part of an integer is the largest square-free divisor of an integer (so the square-free part of $20$ would be $10$), whereas the number theorists assert the square-free part of an integer is the part "left over" after the largest square was removed, i.e., it is the quotient of the integer divided by its largest square factor (making $5$ the square-free part of $20$). [For completeness, what I viewed as the square-free part is referred to by number theorists as the integer's radical.]

With these stories as motivation, do you have a story where a difference in definition for what you thought was "standard terminology" caused a misunderstanding of some sort, whether internally (you personally) or externally (with colleagues)? Stories surrounding notations are also appreciated: my number theory cohorts absolutely hate that I use $\mathbb Z_n$ to refer to the integers modulo $n$, as $\mathbb Z_p$ should be the $p$-adic integers rather than the finite field of size $p$.