Reading through this thread, it seems I disagree with most of you. No surprise there, I suppose.
So, supposing you want to play according to the rules, how should you handle the question posed in the OP?
First, your player has told you that he wants to wait for some sign of casting, then interrupt that spellcast with his readied action. Now, if you clearly understand what he intends to do but still feel that you need to screw with him about whether he used the right terminology, you're doing a bad job of being a DM. Sorry, I know we're all supposed to avoid "badwrongfun" accusations, but the player's supposed to be able to communicate what he wants and you, the DM, are supposed to adjudicate it in game terms. That's your job. In this case, the player has clearly expressed an intent to interrupt the spell being cast, so if you choose to hear that as the trigger being the completion of the spell, you chose poorly.
But how should you rule on the attempt to interrupt the spell? As some have pointed out, there is no specific rule in 5e for interrupting a spell in mid-cast. As a Dungeon Master, though, you don't need a specific rule for every little thing, though, do you? The idea that the lack of a specific rule must mean you can't do something, well... that's just wrong. We've all read the "Actions in Combat" part of the rules, right?
Actions in Combat
When you take your action on your turn, you can take one
of the actions presented here, an action you gained from
your class or a special feature, or an action that you improvise.
Many monsters have action options of their own
in their stat blocks.
When you describe an action not detailed elsewhere in
the rules, the DM tells you whether that action is possible
and what kind of roll you need to make, if any, to determine
success or failure.
The part I bolded is the important bit here. It makes it my job, as the Dungeon Master, to decide whether the improvised action to interrupt a spell being cast is possible, and how to determine whether or not it succeeds. Ok, I can do that. Options include making a concentration check to cast a spell if you take damage during the cast, using disarm to knock a component or focus out of the hand of the caster, or stretching the rule for casting within 5 feet of an opponent to apply disadvantage to a spell attack when the PC throws a spear at the caster during its spellcasting. We have options.
So, moving on to the Arcanist's turn, the mind flayer starts to cast the spell. How should I handle the trigger? This is a fighter, not a spellcaster, so I don't think he'll recognise a spell automatically. I might make an exception if it is one that a party member uses all the time, like ray of frost, but for the most part he's guessing. I know how to do that--passive intelligence (Arcana) to recognise the casting of a spell. Not to determine which spell, just to see if one is being cast in the first place. What should the DC be? Call it 15 minus 2 for each type of component (V, S, M) called for in the spell. If the Arcanist is speaking strange syllables waving its hands in the air while brandishing its corrupted rabbit's foot fetish, you'd have to be a real idiot not to suspect a spell, especially if you're waiting for it. Disadvantage (-5 on the passive check) if the Fighter is looking at the Arcanist in dim light, but maybe advantage (+5 on the passive check) if he's reasonably familiar with what spellcasting looks like and, as I mentioned, he's looking for it. I'll let the player know when he announces he's holding an action that recognising the spellcast will be a passive Arcana check, so he can make choices accordingly (to use inspiration, for example.)
If the spell being cast is a bonus action, I don't think the fighter will be able to do anything regardless of his Arcana passive. Bonus Actions are fast, thus "you hear a sort of grunting burble from within those twitching face tentacles, by by the time you realise it was a spell it's already cast." He can still throw the spear if he wants to.
If the spell is cast as an Action, then it depends on the passive check. Fail, and "the mind flayer seems to be trying to communicate through gestures or speech which you can't understand." He doesn't understand that a spell is being cast, so there is no trigger. I'd let him throw the spear after the spell was cast, assuming the spell had an obvious effect. Succeed on the passive check, and he gets to try to interrupt the cast. What we go with depends on the spell being cast and what he wants to do, and in this case he wants to hit the caster with a thrown spear. The straightforward, uncomplicated resolution is to let him make the attack, deal his damage, and have the Arcanist make a concentration check to release the spell. If, for whatever reason, I wasn't comfortable with that (which for the record, I totally am) I would just apply disadvantage to any ranged spell attack roll, probably regardless of whether the attack hit. If I wanted to get fancy, and there was a material component to the spell, I might ask if the throw was targeting the caster or the spell focus. If the spell focus is disarmed, it might have effects beyond just interrupting the spell being cast as the Arcanist looks for and recovers the dropped focus.
The fighter has given up his Action, possibly a Bonus Action, and is using his Reaction to have
a chance to interrupt this spell. Overpowered, this is not. Nor, in fact, is this at all a nerf to spellcasters. This is an improvised action, made and adjudicated within the rules of D&D 5e. There are probably other ways to handle it, but this one is mine. If it all works, I might give inspiration to the fighter for "thinking outside the box."
You're not the dealer in a casino, or the host of a television game show. Your function is not to just tell players "that's not in the rules, you can't do it." Your job is to run the game, because you're the Dungeon Master. That means if it isn't in the rules, you make it up.
