Good question. The "low on air" to "out of air" conversation took about 5 seconds total. My buddy's regulator was starting to breathe heavy when he gave the low signal. The next breathe he took was very noticeably the last one from an empty tank, at which point he asked for my reg.
Once I had donated, the only priority was to end the dive safely. Different circumstances may have called for more underwater troubleshooting, but that was not necessary here.
If someone gives you low on air, you want to know .. how low... They might mean, as in this case, out of air, but if you start doing other stuff, you really don't know. In addition, you can make inappropriate/erroneous suggestions, (like you did when signaling to continue the dive and head laterally toward the anchor).
Once you donate, the issue is less important, but still worth knowing. For example, it is often required to work the victim's BC on the ascent, because they are too preoccupied. In this kind of situation, it might be useful to know if they even have enough air in the tank to work the power inflator - should that become necessary on ascent or on surface.
Sharing air with someone at 55 feet when there are 3 people in the team is not some huge emergency that requires a race to the surface. I am not suggesting troubleshooting the problem in any significant way, but showing and asking to see a gage is not going to cause a meaningful delay. If they fail to reciprocate and NOT show the gage then I would interpret that as a sign of impending panic and that would be useful to know as well.
If they didn't offer me their gage, I would be offering them a second stage and then try to sneak a peak.
Rather than nitpick the actions taken on this particular dive, I think it is probably more useful to think about what best to do in the future.
But to be honest, I really don't know what any specific training agencies would (or do) say about this particular issue. It is what I would (and have done) in many situations myself.
Any time someone gives me low on air signal, I try to immediately find out what that means.