I think I've read everything WIS has written on this. Dev Chats, Help, and what others have posted as customer support replies in the forum.
It's one of those areas where the sum total of all of it contradicts and is incomplete. Which is consistent with what seems to have been WIS's approach even when WIS employees engaged with HBD. They tried to say as little as possible and let us figure it out.
Here's what I've learned in 20+ seasons -
You may or many not see an increase in ratings when you promote someone. More likely if it's close to the time they would have gotten a bump based on the point in the season (regular season or playoffs).
WIS claims the ratings we see are the only ratings. There are no hidden ratings or numbers that impact the game. So I doubt behind the 55 rating we see the player might be 55.0 or 55.2 or 55.6. And even if they are, that would probably matter in one at bat in one game every 5-10 seasons. So if I'm right or wrong about this, it doesn't matter.
If you believe the ratings we see are the only ratings, then promoting a player does make him better in the short run. If yesterday he had a rating of 55 and I promote him today and the rating goes to 56, I believe he is that much better for the next game. And being that one point better in some categories will matter in one at bat in one game every 5-10 seasons. So if I'm right or wrong about this, it doesn't matter.
Promoting a player in one way or another does not make him better in the long run. I tested always promoting my best prospects before the season ended in the hope I might get a 4-5 point boost over their MinL career. I'm highly confident it doesn't happen.
Unless WIS is just making it all up, or they never wrote the code that worked as they wanted it to when they dreamed up the game, promoting a player does impact his future development, as it puts him with different coaches and in a different level. Generally, promoting a player puts him with better coaches, but I've had more than one season with a turd of a coach and a better coaches at the level below. WIS has also claimed a player should be at the correct level. So promoting him might put him above what they think is the best level, and might work against development. I've never tested putting my #1 draft pick at AAA for 3-4 seasons.
Promoting players does seem to stop them from retiring. So if you're #1 draft pick was 22 when you signed him, don't leave him at Low or HiA at after 2-3 seasons thinking you'll promote him after rollover. He might retire. If he's 24 and you think you might want him in the ML someday, get him to at least AA before the season ends.
By definition, there must be a best possible way to develop a player. That would probably include perfect coaching, which does not exist in the game. I suspect that's why in 99% of the cases players don't hit projections, no matter what you put in ADV. Even if you did everything else exactly as the game want you to, lack of perfect coaching brings the player in short of what was mathematically possible.
As HBD really works it seems that if you invest in coaching and training, put players at a reasonable MinL level, play them at a reasonable position, promote about once a season, rest them when they are tired so they are less likely to get hurt (and back that up with HLT budget), and basically do everything that seems to make sense, you're going to come reasonably close to whatever "perfect" might be. I don't see how it's worth trying to figure it out past that. Probably can't be done. And you'd need mountains of data and test cases to test theories.
The difference between a rating of 60 and 62 is tiny. Maybe one at bat once a season. It can't be the best use of your time to worry about development beyond doing it good enough.