But since they had a stamp, and were flying out, how do the people at the Alaska airline ticket counter know whether or not the visa has been paid for or not? Do they always ask for proof of payment when you don't fly in with them? I've never been asked before....???

Probably if the "entering by" section is checked "air", they assume it was paid by the airline. If it is checked "land", they may request proof of payment.

It's been almost 10 years since I entered Mexico by land from the US, so I don't know what the current procedure is there, but I know for certain that you can enter Mexico from Belize and get your FMM stamped, and drive into the country, without ever stopping at Banjercito. At the crossings here, they check for proof of payment when leaving for Belize. That's why I can say that it is not universally true that a stamped FMM means you have paid the fee, because it is not the case at all crossings. I bet that is just the standard procedure at the busy northern crossings, but not an official regulation. That's just speculation, though.

Leaving for Belize, they'll hit you up for proof of payment of the fee, even if you arrived by air. You need to bring a copy of your itinerary showing the taxes & fees paid, if you wish to avoid "paying twice".

I guess the key take-away from this is that if you're doing anything unusual, such as flying in and leaving by land, or vice versa, then you should keep your receipt/proof-of-payment and be prepared to show it when leaving. But I still don't think it was an illegitimate request or scam.

Your FMM will NOT be stamped until after proof of payment is provided. Period.

That is not universally true. A stamp from immigration is not proof of payment of the fee to Banjercito. Therefore I don't think the request to show proof of payment when departing is illegitimate.

Your FMM will NOT be stamped until after proof of payment is provided. Period.
It's not uncommon for people to get their form, pay for it, and neglect to go back to have it stamped at immigration. Something about understanding the language.....

Back in the '80s in Ecuador, if you forgot that last bit about getting the stamp, you were SOL, and essentially had to re-book your departure and do it all over again. Don't even get me started on what picking up a shoebox-size package from the USA was like!

When you arrive by air, the airline pays the fee.

The airline pays the fee because the airline has included the fee in the price of the ticket. If you have permanent residence, you can request a refund of the fee from the airline.

The obvious argument of "how else would I have gotten this visa" is not valid, because you take it to Banjercito to pay for it after you already have it. If you paid "directly" to immigration in Baja, and did not get a receipt, then maybe that is where you got scammed, not when leaving at the airport.

I've never been to Baja, so I don't know exactly how it is when entering there, but here in Quintana Roo it is entirely possible to enter without stopping at Banjercito to pay, and the enforcement is when turning your tourist card in when you leave. So your situation does not seem unusual, except that they don't necessarily make it obvious what you're supposed to do and keep.

The common "scam" down here is requiring the proof of payment when leaving for Belize, even from air travellers. They are expected to show their full itinerary/receipt from when they arrived, showing the taxes paid, and if not, they will make you pay again. But it is supposed to be a legit payment to Banjercito, though I've read a number of reports of people paying this direct to immigration. That's not how it's supposed to work.