Don't confuse the island league point with the island marker. They're easy to confuse; they're right next to each other like two stars in conjunction, and they both represent the island. However, they mean different things:
The diamond-shaped marker appears on Yer Known World after you "discover" that island. It'll also show up if you've got access to a chart showing it, but you have to actually set foot on dry land to have it on Yer Known World permanently. Being able to see the marker allows you to whisk to that island by swigging on a whisking potion.
The island league point, like other league points, is a blue or brown oval that only shows up if you've got access to a chart it's on (by the way, that means in your inventory or on the navigation table of a ship you're aboard) or if you've got it memorized. It'll appear right next to the diamond-shaped island marker. Having access to it allows you to chart a voyage to that island. Having the diamond marker isn't enough -- you have to have the league point in view. (Similarly, having the league point memorized won't let you whisk to the island if you don't have the diamond-shaped marker available.)
Now, here's the fun point. There are two ways to memorize an island league point. The first, you already know: repeatedly arrive at it while duty navigating at "fine" or better without getting up to any shenanigans during the preceding league.
The second way? You can jump up and down on it.
No, seriously. I am not kidding. Every time you set foot on an island counts as a successful arrival at that league point. And there are so very, very many ways you can set foot on an island.
- Disembark from a ship ported at that island.
- This includes navy ships used for sea-going puzzling missions.
- If it's a crew ship, get back aboard. Disembark again.
- Ferry to the island, if available.
- Ferry to every island you can ferry to. Repeatedly.
- While standing on an island, log out. Log in again.
- A lot.
Requires no badge or subscription. Requires no crew rank. In many cases, requires neither chart nor ship. Barely requires a brain.
So... go jump around a bit, and we'll talk LP memming some more sometime after you're done.