Are Sky Lakes height dependent?

I've created a Large world for farming purposes. What I've discovered is that the Fisher NPC gave me a quest to fish a couple of fish out of Sky Lakes. I'm aware that these are typically naturally-generated as floating islands, but I'm wondering if these must also be in the Space biome to be considered actual Sky Lakes. The natural floating lakes in this world all spawned in the Surface biome (height), and it seems like I cannot fish any Sky Lake specific fish out of them.

Note the biome information at the top-right.

Am I just getting terribly unlucky, or did this spawn too low to count? Can I move it up (with pumps), and if so, how high must it be?

It is possible for the world to produce a sky island with a lake Low enough that it will not allow you to fish out Sky typed fish. I have one in my large world that has this issue, over 30 fish from it and they are all the Forest type of fish.

Seek out one that is another 100 or so blocks up. As mentioned by @Danmaku, if you see harpies you are definitely in the right area but it does not need to say Space.

You won't see a "Floating Island" marker on the depth meter, but the biome is indeed based on height. In some world, there are very high cliffs and mountains where it's possible to be in the sky biome while still safely on the ground.

Floating Islands and Lakes are just the names for the randomly spawned islands floating on cloud blocks; the only requirement for the sky biome is height. Cloud blocks or any other blocks are not necessary to catch Floating Lake fish. You do need to be above a certain height, depending on the size of your world, but not all the way up to space.

The vast majority of Floating Islands and Lakes spawn at the correct height, but some world generation might have one or two islands that were spawned too low and won't count as sky biome. To be sure, look for Harpies: if Harpies spawn, you're in the sky biome.