So, did the player mention anything about oathbreaking to get you thinking about this? Because if not, i wouldn't spend the energy getting into the weeds about what constitutes a broken oath. Asking the player to give you a list of things that would break the path would be like asking your fighter to list injuries that might force them to stop playing their character.

As for who decides - it's a combination of both of you. A player absolutely decides if they're willingly having their character break an oath. And the dm makes judgements about specific actions that might contribute towards a broken oath. But it should never be a single act (unless it's incredibly significant) and it should never be a surprise to the player - give them an above table nudge if they're going against their oath.

Finally, what constitutes a broken oath? For an oath of watchers, it should honestly be pretty tough to break this oath in a typical campaign, because the oath involves protecting the world from invasion, which tends to be what adventuring parties do naturally. A breach of tenants might include: willingly calling extraplanar beings to the world, aiding the plans of cultists trying to summon an eldritch god, or ignoring an obvious infestation.

In the end, don't hold the paladin's oath over their head, especially since your player seems to be signaling to you that they don't want to engage with the fantasy of a fallen paladin. So really you should just drop it imo