First-in first-out

It will depend on what you put in I would guess.
If I put in a Blissey first, and someone comes along and puts in a Pidgey next, then it will come down to the attacker.
The attacker would need to kill my Blissey first - then they can leave, and start again with my Blissey, or instead of leaving they continue to fight the Pidgey next.
I would imagine it will take more fights to fully remove my lvl36 Blissey than it will to remove a lvl4 Pidgey.

So if the attacker goes through the full line up each time - I expect the pidgey will be out first.
...but because they have the option to attack the Blissey first, and then log out and start again on the Blissey, then the attacker gets to choose who will leave the gym first.

First in - first to be battled.
First to lose all motivation - first out.

With some assumptions it can mean first in, first out, but not in gereral.

Until now, we do not know how loss of motivation due to battles or over time and gain of motivation due to berry-feeding will be.

This new 6 mon gym might actually end up being a lot tougher to take down that the old 10 mon gyms.

Granted you can only encounter one Blissey, which is the only truly problematic mon, but imagine if you have to beat a 3k Blissey 10-12 times to take it out. That would be exhausting as you also have 5 other mon to beat a similar amount of times.

So you may end up actually fighting 6 mon 10-12 times each - let's say some 70 fights - to take it down. That's quite a few more than against even tough 10 level gyms. Add to that that the mon you are fighting might regain their motivation from people feeding them berries.

My ultimate fear is that spoofers will get a notification when "their" gyms are being challenged and then they'll just sit on their couches feeding their mon berries while you wear out your entire army multiple times in vain. I wouldn't be even the least bit surprised if that ende up being the case for many players and yet Niantic still refuses to understand that spoofers are killing the game.

EDIT: Of course those calculations are based on assumptions posted by others that a defeat typically leads to a 250-300 point motivation drop.

I agree with you.

It is not fair that the first in loses motivation first.

From what they've said, first in will not always mean first out. Niantic said if a pokemon loses all motivation, it will be returned to it's trainer the next time it is defeated in battle.

So take this scenario:
Gym has a blissey, charizard and vaporeon. The blissey has been fed berries by a few trainers and kept at full motivation. The other 2 haven't. So they're at say 1/4 and 1/2 motivation, respectively.
You battle blissey and beat her, Niantic has shown video that you can now either end the battle or continue to the next defender. So you continue to the charizard. And then to vaporeon.
After defeating those 3, the charizard's motivation is empty.
You battle again and defeat blissey again and continue to charizard. You beat him and you just defeated a defender with no motivation, so it's returned to it's trainer.
Then afterwards, you do the same to vaporeon.

So blissey would be the first in, but in that scenario you kicked out 2 other pokemon first

The shaving issue is the one I'm the most concerned about. It would have been easy to simply use first-in-last-out instead and that would greatly demotivated shavers. With LIFO, they would always end up with their newly placed mon at the most risk of being kicked out due to battling. If instead the first mon in is the first one attacked, then it makes more sense for spoofers/botters to attack a rival gym, empty it and then wait until a few other trainers have put mons into it before putting theirs into the gym. That way they have some buffer between the upcoming attackers and their mons. That isn't a viable option for human beings (who aren't going to want to take down a gym and then sit around for someone else to come by to fill it), so it FIFO benefits the spoofers/botters and hurts the real players.

I'm hoping the information we have is bad or they change this soon.