Basic doubles rotation
Your partner is right about where you move but there is another indicator as to where your partner at the back is hitting it from! He is hitting from where the opponent hit it to!! So you should know even before it goes over your head. Where your partner is. And you can even know where he hit it to quite early because you see the opponents reacting.
If you move to the wrong side after he clears, then your partner at the back has a long distance to move.
I had one funny situation where my partner at the back on one side, cleared it cross court , stayed back, and strolled over to the other side(which he should not have done). I moved back to what would have been the correct side based on where he was when he did his cross court clear. I knew where he was when he did the cc clear so I moved to the other side. But found myself moving backward into him! His fault.
Another thing to learn is which side to move to when you at the front lift it.
The difficult thing is to know soon enough that your partner cleared it .. it’s not difficult to know where he did the shot from(as mentioned). Sometimes if I clear it I will call out “sides” so they know I cleared it and what they should do. As for what side, they at the front should know.
In some cases it won’t matter like if I at the back cleared it from around the centre line so not really to one side or the other.
If it were an attacking clear then I think we would maintain attacking positioning front back.
I remember being told once by a coach what your partner said to you about seeing the trajectory of the shuttle. Though I’m not sure if the purpose of that was to see where your partner was hitting from. I think you can know well in advance where your partner hits from based on where the opponent’s hit it to cos they have to be there if they are to respond to it. The trajectory your partner hits it might be useful to know where he hit it so you can see if you should stay at the front.. if the opponent is under pressure. But also for that. The opponent’s behaviour, responding to the shuttle, might be something you see sooner than the trajectory of the shuttle over you. Anyhow to know which side to move to, (if going sides), you should know even before your partner hits it.
Maybe what your partner meant(but he misspoke) was watch the trajectory of the shuttle coming from the opponent’s, and over your head to him.