I've also been grappling with definitions. I'm 55, now officially diagnosed ADD (ADHD inattentive) and on Strattera which makes a huge difference. It NEVER occurred to me that I had ADD, and I've been involved with mental health issues for decades. I didn't see it apply to me because I can "focus just fine" could pay attention in lectures... I'm not "inattentive". BUT now that I watch myself closer and understand that this is and EXECUTIVE FUNCTION disorder everything makes sense now. Bear in mind also that ADD/ADHD comes with hyper-focus as a symptom. For example right now, I need to start my workday (I'm an IT consultant and work from home, something I COULD NOT HAVE DONE without medication... I'd never get to my desk or if I did, as I am now, I'd never start, I'd do things like "oh I wonder if there's any interesting questions to answer on the ADD forum at psych central, and then, as I am now, get hyper-focused on giving you a thoughtful answer! … oh!! just realized I didn't take my Strattera pill yet today!... seriously

LOL - OK just took it. Hope I can finish this question then start some work... though I really would like to take a walk first... but then would I ever get down to work? isn't it better to get something done so theres some sense of accomplishement then go for a walk? But if I do get down to work I may not want to interrupt myself and I'll never get a walk in... do I need another cup of tea? Oh wait... I'm in the middle of answering a question about ADD...)
So back to the point.... With Executive Function not working correctly It is cripplingly difficult to take anything from thought into action. I see the dishes need doing, I think I should do them now before it gets worse, I see the dishes need doing, I think about other things that need doing, I see the dishes need doing, I think of someone I'd like to phone, I phone them, see the dishes need doing, decide to check my email, see the dishes need doing.... Now with meds, I see the dishes need doing so I get out of my chair and do them!! wow when that first started it was so amazing!
And I think there is something in what you are saying: with executive function not working properly it is very hard to decide WHERE the attention should be, but in addition to that, it is very hard to prioritize tasks because there is such a backlog going on in my mind. We are of course all different. I find it quite easy to put my attention on what others are saying to me, or when doing activities with others, but I'm an extrovert so perhaps that is part of it, I find it hardest times like now when I'm on my own with my own thoughts and I have to direct them in a particular direction without any outside assistance.
I tried to draw my experience once, and I drew a cross section of a skull from above with ping-pong balls bouncing in every direction around the interior, which are the thought and ideas and intentions to act, and a kind of pane of glass behind my eyes through with they could not escape - and me sitting in a big armchair behind the glass looking out, but with the ping-pong balls trapped in there with me unable to get out. It definitely is anxiety provoking.
So yes, in summary - I think the simplified definition and the over use of the word "attention" like you can't pay attention or you can't focus does kind of miss the point in a number of ways, especially since Hyper-Focus is a key feature (whether it's video games , and there are a lot of us who are computer programmers or techs where long intense focus is needed). The problem with the current name, is that for clinicians, teachers,, doctors etc that don't have it they look ONLY for issues of Attention and focus and miss ALL the rest of what makes this so debilitating.
I would like to see ADHD renamed to something that makes this clearer something like "Executive Function Deficit/Disorder with/without hyperactivity"