If you base the logic on orders, then yes indeed. But it could also be that for every ship you sink (eg. every point you drop), some more people talk about it and rumours spread.Black/white is perhaps more realistic. The French Admiralty wouldn't say "Captain Nelson sunk one of our ships today. Instruct all our captains to look a little bit harder for him." But they might one day say "This Captain Nelson has sunk a lot of our ships. Instruct all our captains that hunting him down is a major priority as of today."
The most realistic may be a slow, gradual increase, THEN a huge step and a gradual increase again after that.
That isn't quite as simple and straightforward though
If I'd have to choose myself, I like the gradual one because then every lower number has some measure of meaning in actual gameplay, however small.
The part of that fame that sticks is already measured (indirectly) in wealth.And we're back to historical figures who made a name for themselves by sinking a lot of ships and are still famous today. Fame of that sort should stick. It doesn't need to be nation-dependent. What is nation-dependent is what the nations do based on that fame. Friendly nations love you, enemy nations hate you, neutral nations probably don't care about the military implications but have still heard of you. So friendly nations might do something nice e.g. access to governors' nieces, hostile nations do something nasty e.g. being more likely to recognise you, and neutral nations don't do anything so it doesn't matter.
Would it not make sense to have BOTH mechanisms in place, working together in some way?