Good point about thermal stabilization. I'd add one more spec that nobody thinks to ask for: accuracy under eccentric load.
Every press spec is measured with centered loading. But real dies rarely load the slide dead center. We tested our 300T with a 50% offset load (loading one side of the slide at 60% capacity) and the BDC position shifted 0.03mm compared to centered loading. That's within spec for the press but it means your actual part tolerance budget is tighter than you think.
The way I spec it now: I ask for BDC repeatability at 80% tonnage, at production SPM, with 30% eccentric load, after 30 minutes of warm-up. If the manufacturer can't or won't test under those conditions, that tells you something. Komatsu was willing to do it, Aida pushed back initially but agreed when we insisted.
One more thing — don't forget to spec the measurement method. "±0.01mm" means different things depending on whether they're using a laser interferometer, a dial indicator, or the drive's internal encoder feedback. The encoder will always show better numbers than an external measurement because it doesn't see mechanical compliance in the linkage.