It is very unclear, I recently ran into the same puzzle of how to use the onion correctly without risking loosing it.

this is withg my Bo stabled.
selecting "attend to my chocobo" > train gives the warning message seen in insert #2
this message is really unclear,
yes I know my Bo wont earn exp, yet i can still waste a krakka feed and wait an hour.
selecting "yes" will allow you to select the thav onion, but there's no message or prompt asking you "would you like to raise your Bo's level"
this if anything should be an option that is preminantly greyed out unless
1 - your bo can use an onion AND you have an onion to use.