1. It all depends on the context. Right now people seem to think that counterless art-deco revivals and derivatives possess those qualities. Five years ago they would have just looked dated or historical.
2. Any hi-contrast serif with an appropriate optical weight.
3. That’s why Ogilvy and Pentagram didn’t corner the design market decades ago.
4. Convince them. Or just change and take their money. Or drop them and move on.

How can we best communicate our subjective feelings about typography?

Through typography.

How do you communicate with other people about type?

Here, via email, or preferably in-person over a beer, joint, etc.

Do you reference a particular aesthetic lexicon or classification system…

I tend to go with Vox so people don’t think I’m being pedantic, although I avoid using the term “transitional” because it’s a rather stupid term.

…are your clients on the same page?

I’m pretty sure that when I have clients, they’re paying me so that they don’t have to worry about those pages.