So, the difference I perceive lies primarily in the two bolded sections. Under your method, the player may or may not describe in terms of the fiction what their character says or does (be it in 1st or 3rd person narration, obviously either way is valid and up to player preference), and the DM describes what happens. Under this method, if the player does not describe what the character says or does, then either the DM must do so as part of their description of what happens, or the in-fiction action will be left ambiguous. I see both of these as undesirable outcomes.
Under my method, which I believe to be the one espoused by the 5e rules as written (disregarding for a moment my personal house rule, if you want to call it that, of the player choosing the proficiency), the player must describe in terms of the fiction what the character says or does. This is essential for the DM to be able to adequately determine if what the character says or does has a possibility of success, failure, and consequences, and determine an appropriate ability and difficulty for the check if it has all of the above (as well as what proficiencies might apply.)