In one-dimension this is true; but in 2 or more dimensions it's not always true. Consider the case of two cars passing on the highway - one northbound and the other southbound. They pass with several car widths between them. Let's say each is going 60 mph. At the moment they're adjacent to one another, their relative speed is 120 mph. But their closing speed is 0 mph at that moment.
No. The speed of the cars remains the same; 60 mph each way, so a differential speed, not relative speed of 120mph. If they were 1km apart, there would be no doubt that the difference was 120mph, at 100m the same, and at 1m and so on.
Closing speed applies when the cars are traveling in the same direction.
ETA:
"Closing speed" is often used for closing distance: A car traveling at 20mph, is 10mph faster than one traveling at 10mph, regardless of distance or when passing or not.
Hollywood aside, closing speed refers to collisions, and is:
"The magnitude of the velocity difference at impact is called the closing speed."
The definition, and confusion with differential velocity or speed, has been the subject of a few forum squabbles.
A car traveling in the same direction as an other, may approach and then maintain the same speed for as long as it may, but that can't happen with two cars traveling in opposite directions.
Velocity is relative, so direction is too.
DISCLAIMER: I don't read much of what humber posts, so it's possible your answer to him was right on the money in the given context. I just felt the brief need to converse with someone that can... communicate.
I know you won't respond to my balloon challenge, and you can bluff that out. But, it's a consolation prize, because when directly asked, none of the Carteers were willing to back you, nor did any volunteer to do so.