Unfortunately it does.

A travel day allows unlimited travel on trains scheduled from 000 up to 2359 CET. So if you are on an overnighter tour pass is only valid until you leave that train (whatever time that is). Your next journey requires a new travel day.

Are you starting in the UK, with the Caledonian or Cornwall sleeper?

There’s a quirk here. Everywhere else in Europe, it is the day of the departure that counts for sleepers - so you can travel all day, then board a sleeper in the evening, stay on it until the end, and you’ve only used one travel day (as long as you make no connections after midnight).

In the UK, it is in theory the day of arrival that counts. So boarding a sleeper in Scotland, then taking the Eurostar and further continental trains should only count as one travel day. But in practice, the planner seems to apply the day of departure rule.

It’s never really been cleared up by Eurail what the traveller is supposed to do

Aha!  Yes, the plan is to start on the Caledonian Sleeper, so that’s interesting.  Will bear in mind and look for anything definitive.  Much to learn!

Oh, and there is a secondary ‘quirk’. Midnight, for travel day purposes, is supposed to be midnight CET (2300 UK time).

So the Lowland Sleepers (departure 2340 or 2350 UK time) are for Interrail purposes departing after midnight.

Except again, this policy doesn't seem to be reflected in the app, and needs some clarification.

Correct, we've seen this mentioned before in this community. The app doesn't adhere to the conditions of use, so you'd be eligible to an extra travel day (and home country day) after it uses the incorrect date.