I say "23c" because that's what's printed on the tires. I assumed it was synonymous with 23mm. Is there a difference. And for the record, the only reason I'm still running those tires is because that's what came on my bikes. When they wear out, and by that I mean cord showing, I'll try a 25 or 28c.
It's unfortunate that tire sizes are printed that way - in this case 700X23C. Because the "C" refers to the rim size - 700C - not the tire size which is 23mm. The C attached to the tire size is really meaningless, and the tire size really is 700CX23. I have no idea why this is the conventional nomenclature on tires. A pet peeve of mine, but so ubiquitous nowadays (calling a tire size 23C, 25C, 28C,etc) that I just have to bite my tongue. We pedants will usually reply to such grievous errors by simply replacing 23C with 23mm rather than ***** about it.
But people know what you mean when you write 23C, meaning 23mm.