No, its not fine and I think maybe you misunderstood becuase I wasn't very clear in the first post.

This is a problem of whats painted on the jet, who is operating it, and how ATC sometimes refers to them.

If the controller wants me to follow, give way, turn in front of a particular jet, be very specific. Don't use the mainline operator name only -- especially when there are other aircraft out there. Someone is going to assume the wrong jet and they're going to ding wingtips.

Instead, tell me the exact aircraft type and operator (as to whats painted on the side of the jet).