I think I may be getting a little too precise in my old age, but do you think it’s ok to use the same hand pump for different oils, knowing there’s some residual left in the pump? For example, using it to pump gear oil into a differential, draining it out as much as possible, and then using it to put fluid into a gearbox? Or is the residual gear oil going to completely mess up the gearbox and require me to spend the rest of my life telling people to never use a pump for two different kinds of oil?

I do the "chemistry lab rinse" = drain out the old oil, wipe it as much as possible, put some of the new oil in, rinse around, drain off, fill with the new oil and use Gone to a good home: 2011 4.4 TDV8 Vogue SE Buckingham Blue with Ivory and clear glass = "Rory"
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I'm sure if you sent it to a garage, they would 100% do this all the time.

The residual oil left in a pump is probably less than 20ml, against a device holding a litre or more, if that oil even made its way into the component.

However, I would personally keep a different pump for different purposes I.e I have a hand pump for the gearbox oil, a separate pump for the front and rear diffs and a separate pump for the transfer box then, there can be no chance of cross-contamination (unless you forget to write in permanent marker which is which).

Even better, just buy a new pump for each job - they're only around £10 and the jobs don't need doing that frequently. 2015 Range Rover Autobiography 4.4 SDV8