the 'lithium' refers to the soap type that is used to thicken the grease. The things that differentiate one grease from another include all kinds of other things beyond that, so some lithium greases are OK and others are not.
It isn't uncommon for greases to claim 'corrosion inhibitors' and 'extreme pressure additives', 'excellent water resistance' etc. The reality of these claims is exposed when you find out that there are different (more expensive) greases which are specified for service conditions where these properties are in any way likely to be challenged.
FWIW greases that come in tubes that fit those grease guns vary from 'alright' to 'good' (in my book) but better greases can undoubtedly be sourced without difficulty (for bulk fill guns).
Greases in small tubes work out pricey, too. If you value the convenience, you are not going to use a lot anyway, and the service conditions vs maintenance regime is favourable, this kind of grease gun is OK.
But if not, a bulk fill gun is a better idea. The little dualco grease gun (costs about £12)
is quite handy and is the next step up from the grease guns that take tubes.
cheers
