I did fail to mention that Plumber's Putty be rolled onto the base of the toilet. But this, too, would most likely seal any leak with the addition of the grout as you recommend.
You're right - The toilet shouldn't rock at all. As for the unevenness of tile vs. vinyl, one of the articles you cite says that it's 'debateable' (or still being debated) as to use caulk or grout. So I'll leave that to the Debaters.
Your reasoning is sound. And I used the wrong word 'shim'. I should have said to 'shim with plumbers putty for minor rocking'. (Really, if the toilet is THAT far out of level, I'd say that there are other 'adjustments' that can be made. I don't caulk around the back of the base because of leaks. (Yes, I do want to know about them too.) I've seen caulk discolor, and when I do, I call the mfgr. for a new tube. (Once, I got 4 tubes when I only needed one! I've never had a mfgr. 'deny' me a replacement tube. I simply rip it out and re-caulk.)
The plumbers putty is a very good recommendation. (And I really notice caulk being used because it's a 'smooth finish', and probably because it's 'easier'. It's just what I see and my $.02 as to why I think I see a lot of caulk. I'm not a plumber.) Aaaaanyways, my best to ya whichever 'method/material' you (Melissa) use.