I generally agree with Tess' document to the extent that the party terms are currently not well suited for specification language because of their ambiguity. You're also right that ideally explainers and intent emails should make it easy to discern cross-site vs. cross-origin behavior (seems like a good issue to raise with that explainer).
However, I would be hesitant to forbid or discourage the usage of "party" entirely. It may be used colloquially on purpose, to hand-wave away some details that are provided later or elsewhere. "Party" may also be a defined term in some proposals, like in First-Party Sets. I think there are still acceptable reasons to use the term, and it's so commonly done that blanket-restricting it doesn't seem productive.
I think your email already serves as a good PSA that this is good practice :)