Moving where - who decides?

The laws of chess state:

"4.2      

Provided that he first expresses his intention (for example by saying “j’adoube” or “I adjust”), only the player having the move may adjust one or more pieces on their squares."

Only the player having the move.

When a player has "made" a move by releasing his piece on chessboard, he no longer has the move, right? Even before he has pressed clock, he is not allowed to touch any piece again, and the opponent is allowed to respond...

but there is another rule:

"7.4 If a player displaces one or more pieces, he shall re-establish the correct position in his own time."

in other words, he is allowed to keep touching the displaced pieces, even after having released his own moved piece.

So, what should be done if a player releases a piece at a position on chessboard where it is unclear which of the legal moves was intended? Is the player allowed to adjust the piece to one of the plausible destination squares, while on his own time? Or is he no longer "having the move" after having released the piece, with the result that it is the opponent who has the right to announce adjustment and adjust the piece, possibly to a square other than what the first player intended (although legal and plausible based on the position where the piece was released)?

Heh heh, you said dragon, poison, flagon, & pestle in one sentence.  I saw all those words one day when I played Skyrim.

I can only speak about USCF rules, not FIDE, but a move is not "completed" until after the clock is pushed. Even though a move can be "determined" which means that the player cannot change their choice of move, because of "touch-move" rule and releasing it on the destination square, it is still that player's move (has the move, as in the OP question). For example, you can still lose on time if you don't reach the clock fast enough, even though you have already moved your piece. Unless, of course, it is a checkmate or stalemate position, that ends the game instantly...so a player clearly is allowed to adjust pieces after moving their piece if they haven't pushed the clock yet.

I often witness players saying "adjust" or adjoube" and touching pieces when it is the other players move, which is CLEARLY ILLEGAL.

Just last week I played a League (team) game where my opponent set one of his pieces on the border between 2 squares. I just asked him which one he meant, and he adjusted the piece. I had plenty of time left, so I asked on my own time, but as a US tournament director I would let a player who was in my situation pause the clock while asking.

Edit: to address someting in the original question: "Even before he has pressed clock, he is not allowed to touch any piece again, and the opponent is allowed to respond..." No, the first player's move has not been completed yet, because that player has not pushed the clock. The second player has nothing to gain, except maybe causing confusion, by moving their piece during that time and should not do it until the first has finished the original move by pressing the clock. After that it will take less than 1 second to make the new move and reach the clock, if the second player was ready to do it even before the first player had finished.