I think you’re missing an important scene. After the shot of the glass breaking, and before your other shot of no blood on his hands, there is a scene with Phillip putting direct pressure on his cut with a rag. This is just a small attention to detail Hitchcock provided us so we can see Phillip has addressed his wound.
I cut my hand on a broken glass when I was doing the dishes one time. Based on the water mixing with the blood, and the hand has many small blood vessels, it appeared to be bleeding profusely. After I held a paper towel on it (direct pressure) for a few min, it coagulated, and all that was left was a miniscule line where the cut was (barely noticeable). I always assumed that the same thing happened to Phillip, aside from the liquid being champagne and not water (like in my case).