Well, there does seem to be an underlying sense of that, and you know what? In principle, it's not wrong. In general an action declaration that states both what you're doing and how you're doing it is better than one which doesn't.
To expound on your statement, I'd say "better" in the sense that 1) it gives the player more narrative control over how their PC is acting, 2) helps minimize any ambiguity as to what is occurring in the fiction with regards to said PC, and 3) adds flavor to the story from a participant other than the DM in this shared story-telling experience that we all love.
That said, it's also not a standard to which everyone is willing to adhere every time or even any time. For a lot of players, "I smash the vase" or "I punch the guard in the face" or "I open the door" is more than good enough most of the time.
Sure. More than good enough most of the time for certain playstyles.
Further, there's many degrees of detail involved in the "how you're doing it" piece even when it is given; consider the difference between "I smash the vase with my axe" and "I smash the vase with my axe using a down-angled left-to-right swing at three-quarters strength and employing the Tofflich technique* for my grip."
The detail need not be belabored (unless the player really, really wants to invoke Tofflich, I suppose). The goal and approach method just needs reasonable specificity. I'd even argue that your latter example is
unreasonable specificity, if that is something a DM at some imagined table is requiring. "I smash the vase with my axe" is simple and perfectly fine. The goal? Smashing the vase. The approach? Using their axe. It's a succinct piece of story shared by the player. Now the DM can adjudicate accordingly. But, of course, this has all been said before.
The disputes come around who gets to fill in any details skipped during declaration. For me, if the player doesn't give the details I take it that a) the player isn't concerned with the details and b) the right to fill in those details has been ceded to one or both of me-as-DM or the game rules.
Agreed. At our table, and I'd argue according to the rules/guidance of 5e*, the player is fully in charge of roleplaying their PC (PHB 185: "determining how [their] character thinks, acts, and talks.") The DM is in charge of determining whether said actions of PC succeed, fail, or require some kind of roll. When I get to play, I'm not a fan of the DM narrating my PC's actions (which is very different from narrating the
results of my PC's actions, which is the DM's job.) When I DM, I try my best to avoid any statements that give the impression that I'm controlling/playing the PCs.
* - whatever that is. I just made it up.
* I know you are coming at this from the lens of your modded 1e game, which I appreciate since it provides a helpful vantage point, IMO