Yup, sounds like you got that one right.
If I play advantage on a cautionable challenge, I usually make sure the offender knows I've got it in my sights.
I'll often call advantage, then point to the challenge and say "I'll have that one #6" or even "that one's in the book fella" something similar. That way the player has been cautioned in a sense, and anything he does after that is a second offence.
Never had an issue with it, except once when I'd said that, play went on....and on....and on......eventually stopping for a corner at other end of the FOP, ball had sailed miles away over the bar. The player who I was going to book went off to retrieve the ball, jogged off, had to climb a small tree to shake it loose, jogged back, placed the ball ion the ball in the quadrant and took a quick one short. He hadn't actually re-entered the FOP yet, so I had to blow up, pull it back to the corner and issue the card. I had made my notes of course while the player was up the tree!
Also, there was an appeal against a card (another referee) by a player sent-off for 2nd bookable. Ref had issued first yellow then 20 minutes later player commits another offence. Ref takes name, notes down properly in book, then pulls out red card, shows it and walks away. Realising what he has done, he turns around calls the player (who is now near the touchline) to turn around, and shows the second yellow.
In your example I am interested that you write the BLUE captain was arguing that you should have shown 2nd Y to his own player.
We are told that the caution happens as soon as you tell the player - and that the card is to notify everyone else.