Yes, sunset is when we SEE the last bit of the sun disappear below the horizon.
That's true even though, because of the time it takes light to travel here from the sun, and because our atmosphere bends the light and makes the sun appear slightly higher in the sky than it is at that time of day, at the time we see the sun drop below the horizon, the sun has already been below it for a while.
That nice reddish disk of the sun we see at sunset is actually an illusion, a mirage caused by our atmosphere. If you were watching the sun go down and at the halfway point someone suddenly whipped the atmosphere away, you'd see the sun setting one moment and no sun the next, because the sun is already below the horizon when we are witnessing it go down. That's why sunset is when we SEE the last bit of the sun disappear below the horizon regardless of where the sun actually is in relation to whoever is watching it set.