We ran into some issues similar to those that Logdotzip and his friend encounter in their video. In that video (starting at 4:31), they instantaneously "switch to the old classic version", but don't explain how they do it.
I think they are switching to Minecraft 1.1, which is indicated on the puzzle map page:
This map was made in Minecraft 1.1, it's important to play on the same Minecraft client version. Learn how to change your minecraft version in our guide.
That guide explains how to set up a new profile in the launcher, and then relaunch Minecraft. But in the video, they appear make this change instantaneously – as if they can do it with a key press. Perhaps there is an edit in the video which cuts out them both restarting Minecraft and navigating back to the same position?
Minecraft 1.1 doesn't seem to have the
Open to LANfeature, which my son and I use (we are playing on two Macs in the same house). In the video, perhaps they are using a separately running Minecraft server?
If what I've written here is true, then perhaps the way to get this working in our environment is to use Minecraft 1.1 and run a separate server. Or perhaps find the earliest Minecraft version that includes
Open to LAN and try that with this puzzle map.
Does this sound right? Any advice appreciated.