Am I damaging my turntable?

You should definitely NOT do that with your Rega, but there is an easy solution. The felt mat on the Rega will allow the record to slip, so with the platter rotating you can just grab the record with both hands by the outer edge and lift it off the platter. You can then place the next record over the rotating spindle and drop it on the platter and you're ready to go!

Definitely no on a belt drive table. I would also avoid doing this on the SL-1200, unless you use a felt slip mat, which will allow the platter to turn while the record stays in place.

It is easy to pick up a spinning record off the turntable.
I am 64 and can barely pick up a spoon.. yet manage to slip the lps off and on the spinning platters of both my Kuzma and Rega.

The Sl-1200 has a brake. Hit the button and turn it over!