Clearly, the move must be taken back and the clocks adjusted, does this count as if an illegal move also?
No. There was no illegal move.
The FIDE Laws of Chess define illegal moves and positions so:
3.10.1 A move is legal when all the relevant requirements of Articles 3.1 – 3.9 have been fulfilled.
3.10.2 A move is illegal when it fails to meet the relevant requirements of Articles 3.1 – 3.9.
3.10.3 A position is illegal when it cannot have been reached by any series of legal moves
What you describe is a touch-move violation as described in 4.3 Also relevant is 4.2
4.2 Adjusting the pieces or other physical contact with a piece:
4.2.1 Only the player having the move may adjust one or more pieces on their squares, provided that he/she first expresses his/her intention (for example by saying “j’adoube” or “I adjust”).
4.2.2 Any other physical contact with a piece, except for clearly accidental contact, shall be considered to be intent.
4.3 Except as provided in Article 4.2.1, if the player having the move touches on the chessboard, with the intention of moving or capturing:
4.3.1 one or more of his/her own pieces, he/she must move the first piece touched that can be moved.
4.3.2 one or more of his/her opponent’s pieces, he/she must capture the first piece touched that can be captured.
4.3.3 one or more pieces of each colour, he/she must capture the first touched opponent’s piece with his/her first touched piece or, if this is illegal, move or capture the first piece touched that can be moved or captured. If it is unclear whether the player’s own piece or his/her opponent’s piece was touched first, the player’s own piece shall be considered to have been touched before his/her opponent’s.
What you describe falls under 4.3.1
Regarding your question:
one moves another piece (in an otherwise legal way), presses the clock, and the arbiter observes this; how should the arbiter decide?
The arbiter should decide that the touch-move rule in 4.3.1 has been violated and as per 4.3.1 "he/she must move the first piece touched that can be moved"
The arbiter should then select one of the penalties in 12.9.
For the first such offence the most lenient, a warning along with an explanation of the rule, would normally be appropriate. Even if the offence were repeated repeated warnings might be in order, for instance for young inexperienced players. For more experienced players who would be expected to know the rules a time penalty may be appropriate even for a first offence in light of the disturbance to the opponent.