Here's the history, at least for the US.
When I hired on, way back in 1985, the FAA had printers based on IBM Selectric mechanisms. They had a two-color ribbon, red and black, and a typeball with capital letters in two sizes, plus a few special symbols.
When a strip was printed, color could be selected by the computer for certain items. Callsigns on eastbound aircraft, for instance, were printed in red.
A red route appeared when an aircraft filed a non-preferred route to or from certain airports.
In the US, red is used for a PLANNED route or restriction - something the controller SHOULD do, but hasn't accomplished yet. As an example, if you filed KBFF..KDEN in your C650, the strip would reflect your filed route, but the computer would include a route over LANDR, like this - "KBFF.LANDR1.KDEN" (routing would be over AALLE, today)
That route would be printed in red, to alert the controller, and was the forwarded route. In other words, all subsequent positions would show that route in black, with no reference to the original filed route.
The controller issuing the clearance is expected to use the red coordinated route. If they choose not to (such as for a MEDEVAC), they must update the flight plan to force processing of the original route.
Today, there are no "red routes" per se, if you observe a preplanned route in EDST, it's highlighted in blue. Our strips no longer have two-color capability, so "red" text is now printed in black with a grayscale highlight.
The jargon, however, lives on!
To further address your example, if both a red route and black route exist in the system for a particular aircraft, that means the preferred route has NOT been issued, and some action must be taken. The controller with responsibility for meeting the restriction for that route will issue it, or force the computer to override the red route. Either way, it ensures that subsequent sectors get the right information.