I come from the Old School, AD&D days when paladins were very tight as a class.  They had the highest standards for AS just to create them, and they were expected to be played like Sir Galahad, very very noble (although he did have that one weakness ...).

Fast forward to 2020 and I am playing 5e and we have two paladins in my party.  One tries to play very strict on the noble rules and says stuff like, "If the Rogue tells me he is going to do this then I may have to help the town catch him."  And our other paladin says often, "I'm Lawful Good, not Lawful Stupid."

The first one can be a kill-joy but we can work around it.  I find myself very annoyed with the second paladin who refuses to come to the aid of the town until he sees we have the upper hand in the fight.  And he ignores many ethically difficult questions by just ignoring there is an ethical issue on the table.

Is this second sort of paladin common in the days of 5e?  Is the first sort of paladin common in 5e?  Is this a situation where I should just let the players and the DM agree on the boundaries for playing this character and let it go?  My guess is the answer to my third question is "Yes, let the player and the DM discuss the boundaries for playing a paladin.  It's not YOUR character."

I remember those 1e days when you basically had to cheat to have the stats for one. Paladin ethics have been sliding since 3e when they opened it up to any alignment. In this edition especially since they toned down the divinity aspect to the point where it doesn’t exist anymore. 
From an RP standpoint all a pally has to do is follow their oaths. Depending on the type of paladin this is, the player may be doing it exactly right. 
So first, I would consider that, are they following their oaths. The answer to that can guide you. You could maybe get by with an in-character, “aren’t you supposed to do X right now” but I’d be very careful about it. If they are just meta gaming and taking all the class benefits with none of the restrictions, that’s a bit problematic. But if they are just playing the game and having fun and no one is getting hurt, I’d say live and let live. 

Paladins don't have the uber-restrictive code of conduct they had in 2nd or 3rd edition, but they still have some standards of behavior that they're expected to meet, and none of the Oaths really condone cowardice in the face of the enemy.  You're within your prerogative as a GM to issue a warning regarding lack of adherence to their oath if you feel it's appropriate, and to further zing them with alignment shifts and potentially even falling to Oathbreaker status if they continue.

Quote:

 If they are just meta gaming and taking all the class benefits with none of the restrictions, that’s a bit problematic.

That's pretty much how I see it.

I still think my only choice is to stay out of it.  Thanks for confirming that things have changed, but they haven't changed that much.