What exactly is a "gentleman farmer"

in the context of 18th-century France? Is "landowner" anything like a synonym?

Yes, although in the UK the term gentleman meant a little more. To be called a gentleman you would need to be:

- male, obviously
- landowning, or at least to have sufficient private income or funds not to have to work.
- respectable. This would involve observing the rules of polite behaviour, not abusing your position of power, etc.

For an example from Pride and Prejudice, the hero Darcy, having land, and caring for his unfortunate relatives, is a gentleman. Mr Bennet, having no land but a small private income and is quietly studious, also qualifies. Wickham, the villian, is from a wealthy family but, by seducing a vulnerable wench, doesn't live up to the moral code of decency and is therefore 'no gentleman'.

Dictionary.com provides the following:

gentleman farmer
n. pl. gentlemen farmers
A man of independent means who farms chiefly for pleasure rather than income.