Using the colour of the grease to determine the type is going to be as effective as trying to figuring which fruit someone owns based solely on the description of 'it's round'...
I've seen bearing greases that're red, blue, black, dark brown, light brown and clear - and possibly more that I can't remember. I've seen both blue and red grease that turns black when it's been used, you couldn't tell that it started life as a different colour.
So, that means that it's necessary to either find out what the correct type is or work out what's best, rather than taking wild guesses.
I can't see why rubber grease would be suitable for a puller, a good quality general purpose grease seems appropriate in my mind. Longlife complex, high temperature stuff from Morris never seems to let me down.
Choosing the grease by colour was tongue in cheek, I'm well aware that the colour in grease is just whatever dye the maker has chosen to add. Like you I have used all colours of bearing grease & it differs by manufacturer apart from food safe grease, it's usually white (and useless as a lubricant ).
I do happen to have some (light brown ) castrol longlife high temp grease in the gun at the moment, that might be worth a try. I also have a little bit of a high quality rocol (blue ) bearing grease which should be better as it's a bit thicker and from what I recall it was quite expensive!
But thanks for your advice, it will educate others in the world of greases!