A variation on your variation: A couple of the busier airports I've been to, it's part of the taxi instructions rather than the ATIS. For example:
- "Bigplace ground, Bugsmasher 123ab, at fuel island with ATIS info Delta, request taxi to RWY10, VFR departure to the east."
- "Smasher 3ab, Bigplace ground, taxi 10 via Charlie, Alpha, cross 26 at Alpha, advise run-up complete."
I've always interpreted "advise run-up complete" as meaning "don't go over to tower until I say so". Later on, it's:
- "BP ground, 3ab, run-up complete"
- "3ab, BP ground, contact tower, advise them you are VFR"
From there, I taxi from run-up to the hold-short line switching over to tower somewhere along the way. However, sometimes they reply "3ab, BP ground, hold position, contact tower" or the dreaded "standby".
Another funny thing is that a few times, they tell the folks in front of me to "advise run-up complete", skip that instruction for me, and then tell the folks behind me to "advise run-up complete". In that case, I just switch over to tower in the blind after my run-up.