I don't think the word "layover" has specific meanings but the word "stopover" has very specific meaning in the airline ticketing/fare rule world.

In the old days with paper coupons, a circle in front of a city means it is a stopover (X means not a stopover). Stopover means you have a through fare and it allows 1 or more stops for 0 or very little additional fee. The stop can be as long as the validity of the ticket.

A stopover does not occur unless you stop in a place for longer than the connecting time allowed by the fare - I guess domestic is 4 hours nowadays and international is 24.

My understanding is that airlines have different policies on luggage. For example, if you do LAS-SFO-NRT and end up overnighting in SFO with a 23 hour gap (not a stopover), I think UA will cheeck your bag to SFO since they have a rule on how long they will have the bag at a transitting point.

Another exception of course is the "clear customs at the first entry point" to many countries/regions. In most other circumstances, no stopover ticketed = bags checked through. It seems it is hard to get them to short-check bags.

Going back to your MP question, I think it is safer to have them ticket with a stopover. Depending on the ariport, you might have to pay some extra taxes for it. What I don't know is that if the website can do it if you don't stop for more than 24 hours...

Also, if you have no status, think about the baggage fee complications you might encounter...