I use synthetic brake caliper grease. It's the one grease that I buy that's different from my usual mobil 1 synthetic grease that's used on everything else. The one on the left you have, the hi temp stuff, would probably be ok. I think the theory is that the hi temp stuff avoids "melting" and leaving the area where it's needed, and not traveling to areas like pads and rotors where you don't want grease.