Clerics have their source in faith, and might be required to uphold their faith by actions; in fact, an old way to differentiate them has always been to require a Cleric to act according to their faith or temporarily lose the ability to cast spells. It's not nice to use it as a punishment however (like the DM saying "ha! you just did something opposite to your dogma, you lose all your spells!"), instead it should be the player's initiative to roleplay her Cleric PC properly, not pick e.g. the Life domain because it's powerful and then roleplay a murderhobo.
We've seen how this works out for the last 30+ years though, and what actually happens, is two things:
1) Gods who have potentially onerous restrictions on behaviour for adventurers just do not get chosen by players, or on the rare occasions that they do, last like two adventures before being retired or whatever. No, that one time someone played an SP of Illmater or whatever, and against all odds, it worked, doesn't mean the general trend isn't a big issue.
2) Loads of domains are powerful, and any gods with "adventurer-friendly" tenets will get played, and the players won't even have to alter their behaviour, because, in old V:tM terms, they picked "The Path of What I Was Going To Do Anyway". So you effectively just limit the pool of domains to "whatever domains are possessed by adventuring-friendly gods".
And this is essentially an RP restriction on a single class, which is gains nothing for it. Which isn't great.
It also goes limits what the
DM can do with enemy religious casters very steeply! You can't ever have a "religious leader gone bad" scenario if you work on a "you don't get spells if you aren't a good boy" deal, because you can effectively detect any divine spellcaster who isn't following the tenets of their religion by simply asking them to cast a spell.
I think the 4E (and arguably 5E default) approach that when you become a Cleric, you are granted a divine spark, and your power comes from that, is much more interesting and retains much more flexibility for the DM. That way, you can have clerics "go bad" or the like, and it not be trivial to deal with them. You can also have the situation where the god knows his cleric has gone bad, and sends you after him - he doesn't want that divine spark being used for evil. You can still have Fallen Paladins and the like, too - they are oathbreakers, and that likely causes divine retribution.
As for the OP, I don't think it's a useful distinction in 5E. I don't think it has been for a long time, really. Psionics only being fouled up by anti-magic fields isn't a big deal, because it's just a spell in 5E, and one could always say it was actually an "anti-magic and psionics field" and then you never have to think about it again. It's useful to separate the CHARACTERS and the origins of their power, it's not useful to separate the magic results of that power. Especially when Arcane Casters like Bards can cast healing spells and so on (which has been the case since 3E).