I feel you on the players throwing a fit during the session. Is this going to kill your character? No? Then please just let me make a ruling now and we can discuss it after game.
As far as the distinction in 5e (which, as we've seen, a lot of us unfortunately won't even catch unless we see online discussions about it) I think there is merit to its existence. Having been working on designing a lot of spells recently, sometimes it's subtle distinctions that determine how well (as in balanced and effective at working as intended rather than some other way) a spell is.
Because I have a lot of 5e experience (both actual play and speculative) I find myself refining and carefully rewording spells to attempt to make them work as intended.
Had a spell that makes a 5' aura and you have to save or get pushed back. Then I realized someone could just keep moving forward until they make their save or run out of movement. It might be clear to me that the intent was if you fail you just can't get into that space on your turn, but that is not what it said.
So yeah, since the book doesn't have a section that officially describes the difference between move into and enter (like it does for describing attack = make an attack roll) they really should have put more words in the spell. Something like "voluntarily uses their movement speed to enter the area". However, I would suggest against summarily treating all forms of getting into an aura the same, either by going with the hidden intended jargon (frustrations aside) or by carefully examining the spell to determine why they might be distinguishing between those things (as well as start of turn versus end of turn damage). While sometimes they goofed, most of the times those aspects of the spell really do function best when used as intended (once we know how to figure out what that is!) (I'm addressing 2014, no clue how it works in practice in 2024.)
The player's main beef was how he got pushed on the enemy's turn, took damage, then at the start of his turn, he'd take more damage and drop without having any real way to act. Which I get- those kinds of moments just suck, which is exactly why my group has so many reaction abilities (enough to make me dread attacking them, lol).
But I don't see how poking holes in spell effects to elude them because they were written in a less than clear manner or because someone made an off the cuff remark "clarifying" something without issuing direct errata in newer versions of the rules to muddy the issue bother me.
I mean, they've employed tactics like these before without a quibble and I just accepted it ("yes yes, your eldritch blast pushes someone into the moonbeam, they take damage"). Never before did anyone say "wait, he wasn't
moved there!".
What bothered me the most about it was when I said "look, you obviously feel strongly about this, so ok, we'll play it this way from now on, you only take damage at the start of your turn", but the player was like "no no, I'll take the damage." And when he was pulled out of the spell area, he was like "ok, I die then because of the
booming blade effect" and when I said no, that spell says the movement has to be willing, he as like "well, how should I know?".
I mean, you took the time between turns to look up
sickening radiance and found some forced movement discussion that you cited to tell me I'm wrong...but didn't bother to look up this other spell?
Then he started saying "well if the rules work the way you say, then people should take damage when the Cleric approaches them with
spirit guardians" and I was like, "no, that's been clarified. There's a difference between moving into s
pirit guardians, and having spirit guardians dropped on top of you". To which he said "that doesn't make any sense".
"I didn't design this bloody game!" I finally said, throwing my hands up in the air. "It's a balance decision, not one that makes sense."
"Well, then, it shouldn't have to make sense how I don't get burned by being pushed into the spell."
And when someone else was rescued from the
sickening radiance before their turn, taking no damage, which makes even
less sense, lol, I could feel him glaring at me. He didn't talk much on the ride home either.
And I kept thinking that this all could have been avoided if the game's rules were clearer (or that Crawford was- this is far from the first time I've tried running the game by the book only to have some comment of his thrown at me, lol).
That irritation is what led to the OP. So far from what's been said by everyone here, I was technically correct...but I'm not going to say right, because maybe this sort of thing is kind of BS, but I can't help but think that this is what
sickening radiance is for, since unless you can trap enemies inside, casting it leads to one round of damage, then everyone runs out, and the fact it lasts for a whopping
ten minutes becomes somewhat meaningless, but I'll probably never look at the spell again because of this incident!
For me, having my player's trust is important as a DM. If they feel I'm being unfair, then I guess I have to switch tactics. This entire last battle, I took the opportunity to turn their tactics against them, which was cathartic to a degree, but given that it made the game less fun for them, I'll have to not do this in the future.
I don't see D&D as a "me vs. them" exercise, but I don't think they realize how annoying some of their shenanigans are, lol.