The Injector is really just a clever pump using temperature and pressure differential, but a few locos in the UK, and more overseas, used steam-driven mechanical pumps ("feed water pumps", above) to put the water in.
For example, the Glasgow & South Western Railway had a batch of 0-6-0s built around 1900 with these, invariably known as "the Pumpers" as anyone who had read David L Smith's books about the G&SW will know. They were used because they had another technical innovation that didn't quite make it, the pre-heating of water in the tender with exhaust steam piped backwards. Injectors depend on temperature differential, and don't work well with hot water coming in. Another use of feed water pumps therefore is condensing locomotives, where the exhaust steam is condensed and used again, just like what was standard practice on steam ships (which all used feed water pumps rather than injectors).
Great Western locomotives, and presumably many others, had two injectors, one which worked off boiler steam, which used quite a lot of steam, and one which worked off exhaust steam when the loco was running. The live steam injector, when switched on, is the device which is making the "singing" noise when a steam loco is standing.
Some pumps especially in the early Victorian era had pumps worked off the motion
An opposite sort of pump is the vacuum brake "ejector", which sucks the air out of the vacuum brake pipe. GWR locomotives also had two of these, one large steam operated one which you use to get the brakes off to start, and a smaller one driven off the motion, used pretty much all the time when running just to maintain the vacuum (it was attached to the crosshead behind one cylinder). I believe the latter was unusual on other railways; it is what can be heard making the "tiff-tiff-tiff" sound when a GW locomotive is coasting.
Certainly never heard of greasing the rails to get the latter operating! Mr Churchward would have had a fit. The nearest we came to this at Taunton shed was the coaling stage, which had about a 1 in 10 ramp up to the upper level (there's still one such at Didcot museum), which a Pannier Tank could just about manage, with a short running start, propelling one or two loaded coal wagons, but if it was raining heavily they might well slip substantially, Some shed turn drivers were better at this than others, and just about make it over the top at the first attempt, with huge wheelspin, sparks, and a thundering exhaust noise that could be heard across half the town.