Is it better to replace this flange or install a “repair kit”?
It looks like the flange might be screwed down (note the screws in the flange itself).
The repair flange you linked is a &quot;screw-down&quot; type where you have a raised flange and want to secure it to the subfloor. I'm not sure I'd use that type here (a tensioned &quot;push-in&quot; type would be easier) but I'm unconvinced you need to replace it.
The situation where you need to replace your flange is when the narrow part (where the flange bolts grip underneath to secure the toilet) is rusted out and no longer holds the flange bolts. This flange looks older, but it's metal and the narrow parts are intact. If you can fit new flange bolts on either side then you're fine to put a new wax ring in and re-mount it.
Why is your toilet moving then? Likely the floor isn't level and your toilet can move on the flange. Here's what I would do

Buy a new set of flange bolts and try to re-mount the toilet (dry fit it without the ring).
Once you've tightened as much as you dare, see if it still moves. Look at the base of the toilet for gaps. If you find gaps, add plastic toilet shims until it doesn't rock
If you still can't fix it with shims, then and only then would I consider adding a new flange. Start with a rubber tension replacement flange, since these are relatively easy to install