My OW instructor was not super clear on the fine points of buddy checks
Basically (and without being long winded) the finer points are that at each stage, the diver checks and assurese themselves that each part of their kit both functions correctly, is rigged correctly and they can reach/find it. For the Buddy, its so they are aware of the divers kit , function and placement and the buddy is a second set of eyes. I always teach the for releases, the buddy physically checks the cylinder band is tight and the cylinder is secure.
Checking tank valves is a difficult one. There is always the risk of someone turning it the wrong way (OFF) equally I don't mind being checked because I know that I myself am fallible
Without being critical, because you did well and I note from yoru profile you haven't taken Rescue. but some pointers to think about
If you remember your OOA air sharing skill. On signal of OOA by a diver you should hand them your Alternate (or primary depending on your config), rather than the OOA diver trying to grab/find it. At teh OOA or Low on air signal, you, the donating diver takes control
On teh ascent you should have positive contact, holding either their arm or BCD with yoru right hand. This leaves yoru left hand to control the ascent (by deflating your BCD and checking your gas contents.
Remember once the other diver has a working air source the emergency is over (assuming you have sufficient gas) While instinct does take over , there's no reason why you can't make a SS, and you can signal the other diver to hold and show them yoru SPG so they know they have gas (Indeed letting them see your SPG does give them some reassurance)
I advise all of my divers to keep their skills refreshed in their mind, either by physical practice or by reviewing them online (may good Youtube videos) and then mentally practising them