Back to the OP’s original question… ‘Normal’ grease is, in fact, lithium grease: it’s by far the commonest type. Sodium-soap-based grease is very rare and has, as far as I know, no properties useful to us. Calcium-soap-based greases have better water-resistance than Li grease, which is poor.
As has been said, ‘grease’ now covers an alarming multitude of formulations – quite beyond a mortal’s ken. ‘Channelling’ greases are now popular, formulated to squeeze out of the way of balls and rollers in bearings, to reduce grease churning and heat production. I think they get this property from their fillers (of which there’s a bewildering choice). These behave differently from the softer greases of old, and I’m frustrated by not being able to find equivalents (at least in NZ). The NLGI classification relates to something like viscosity, but there’s far more to a grease’s characteristics than NLGI1, NLGI2, etc, despite what the retailer would have you believe…
Decades ago, Castrol LM grease (a good, multi-purpose Li grease) was a soft, translucent formulation. It tended to separate fairly readily, so there was a thin oil layer on top of the grease in the tin. Whatever happened to it? Now, a ‘multi-purpose’ grease seems to be a thick, sticky, opaque mess, with a higher filler content, presumably with marked ‘chanelling’ properties. It doesn’t separate out in the tin so readily, but, if left in a grease gun with a spring-pressurised resevoir, the oil dribbles out, eventually leaving a very stiff cake in the resevoir, which is the devil to clean (been there…).
My belief is that ‘chanelling’ greases are not good in slow-moving plain or sliding bearings, for which the ‘softer’ greases were OK. Can someone enlighten?