Well I have bought many types of the so-called helicoid grease and sad to report NON have the properties of the OEM zeiss grease
Some of this below echos others:
I occasionally use a helicoid grease some places on microscopes, but not often. It is Helimax-XP
("optical and Instr. helicoid grease w/ lithium & PTFE white- NLGI#1"), and I now only use it on multi-threaded focusing eyepiece tubes (also good there on telescopes). A thin grease, but 'holds its shape'. Used a lot by camera lens repairmen. Has to be thin since the mating surfaces can have many square inches/cm of contact area.
For lubing objectives, a consideration is the volatility of the grease (mass-loss over time), since the volatiles may plate out on your optical surfaces. I've taken to using DuPont Krytox GPL-207
on the helices of objective internals, it's a synthetic with extremely low outgassing.
1) the old grease might have hardened up considerably in your old scope, so you may want to replace it with something that is not so much like the tar you're cleaning off.
2) as many have pointed out here, the consistency is important; the final operational 'feel' you want for that particular part determines the grease.
3) many parts could actually benefit by lubing with oil, rather than grease; this is judgment call. In many cases, grease is used because it is less messy during factory assembly, and/or stays in place better.
4) open gears & racks & ball-bearing races (like in many stages where you can actually see the bearings & races) probably should not be lubed at all, since this holds grit, dirt, and dust in the mechanism necessitating more frequent cleanings. Just lube the pinion shafts with some light oil.
5) dovetails are a special exception to 4), either oil or grease can be used in a thin film to get the sliding action desired.
6) if a non-optical sliding part that you know is supposed to move, doesn't, try seeping a little naptha into the cracks & let sit for a half hour and try moving it again. Repeat this until you can get it to move; when you think the naptha has dissipated, seep some of a 50-50 mixture of naptha and your favorite light synthetic oil into the cracks & work it back and forth. This may make the part operational until you can get time to dismantle it for cleaning (which may turn into forever...lol).