That's a fair way to handle it, but not the only way. And, I'd argue, not quite the way 5e was designed for, ideally. You certainly can run 5e with above-board formal house rules. You might find you need a lot of 'em. IMHO, running 5e on "rulings not rules" works even better. You do lose the implied consistency of formally changing the rules, but you're not boxed in by those changes, either.
But being "boxed in" is a feature. What it says is that the DM has communicated clearly to the players so they know what to expect. I'm all for the DM making rulings, especially in a game like 5e. Even ones that differ from the written rules, not just clearing out ambiguity. But if you change from the written rules, tell the players and stay consistent. That's what a house rule is. I find it hard to argue against either of those.
BTW, "stay consistant" doesn't mean that if a ruling (either in the books or by the DM) doesn't work it can't be changed. Goes back to communicating. "I know last week I said X, but Pat brought up that could really be abused so I'm making it Y." That's fine.
It may well be that there is a good reason for the DM ruling differently at different times. (Of course, there could be a 'bad reason,' that's why the afore-mentioned trust is so important.)
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If by "good reason" you mean ruling a different way in different circumstances, sure. If you mean "the same thing acts differently for some players than others", "acts differently for foes and players" or "acts differently over time without notice", then I would have to disagree. Again, if something is broken than fix it. But otherwise the rules are supposed to be a shared understanding of the mechanics.
There's no need to 'prove' or even argue that RAW goes one way or another, for instance.
If I sit down at my FLGS to play an AL game and the DM that week tells me that an 18 Str doesn't add to attack with my longsword because the DM feels like dex should control all to-hit, then yes, there is a definite need to "prove or argue RAW". On the other hand, if a DM started a campaign and told us ahead of creating characters "all weapons use DEX to hit and STR to damage", then that's a different story.
The rules are there for all of the people playing, not just the DM. That's why we chose a game system. We can RP and tell collaborative stories no matter what rule-set or even none, but we pick rule-sets the table likes and having that mechanical framework adds to the experience.