Yes there are lots of contradictions, maybe this is due to a 10-15 second glimpse, of a yet to be identified aircraft.
I don't have any problem with your inability to ID the aircraft, I probably couldn't have either, based on such a brief glimpse.
But then nor would have I expected anyone else to come up with an identity if I was only able to give them a vague description, and no information at all as to timings.
If it helps, your current theory of a Gulfstream going into Luton is as likely as any - GLF4s/GLF5s are a common sight there, but that unfortunately means you are unlikely to be able to pin it down to a specific movement.