Minecraft Classic saves the map 3 seconds after a block has been edited. It then tries to load that same map if you visit classic.minecraft.net again. This means that you can make a "legit" run, showing the start screen, in a map that you've changed to suit your needs.

Since the spawnpoint is picked randomly between (32,32) and (96,96), it might take a few tries before you spawn where you want, but you can easily get <0.2s times by cheating like this.

Simply require people to show that they clicked the "Generate new level..." menu in their videos.

Generated levels until I found one with a tall mountain, manually built the mountain even higher using dirt, manually added a tree on top, with the top of the tree just below the build limit. Reloaded the page until I spawned on top, got a time of 0.118s.

Generated levels until I found one with exposed iron in the right direction, manually added a beach in front of it so I had as many spawn points as possible that would look directly at it. Reloaded the page until I spawned in front of it, got a time of 0.001s.

I got a bit suspicious of the border% records due to the gap between #2 and #3, so I decided to look into the code that runs Minecraft Classic (web).

TL;DR: It is impossible to spawn less than 32 blocks from the border without cheating. The #1 and #2 times on border% are cheated

The code that picks the spawn location is:

Since this is obviously minified, and thus fairly unreadable, I took the liberty of translating it into pseudocode:

In other words, the spawn point is chosen randomly within (worldsize/4) of the map center. For this reason it's impossible to spawn at the border.

Combined with the fact that both #1 and #2 don't show clicking the "Generate new level..." menu, and start at the pause screen, I think it's fair to say they're both cheated.

The #3 time, which shows the "Generate new level..." menu, spawns exactly 32 blocks from the border, and is thus close to perfect.