Got to use the tent that fits under the canopy for the first time last weekend. Worked great.
As you can see the tent, which is meant for a 10x10 straight-leg canopy, fits well inside my 12x12. Look closely and you can see how I've made that work by rigging loops of paracord for all of the tent's attachments.
The dirt/gravel in the background is very hard (doesn't take well to stakes) and also covered with sharp stones. The other style that hangs off the edge of the canopy, and which requires staking, wouldn't work well under these conditions.
Setup and takedown were both very easy and I still had half the canopy footprint available.
I did put down an 8x10 heavy-duty tarp first to protect the tent floor from water, dirt and unnecessary wear. I expect I'll be replacing that tarp from time to time.