The Red Cross, Red Crescent, and new Red Crystal are specific logos (like all trademarked logos), with the additional aspect that they are recognized in international laws, including the Geneva Conventions. Vehicles, for example, carrying the Red Cross logos are not supposed to be fired upon (
https://www.haaretz.com/1.4929066), and no other vehicles are supposed to display the Red Cross logos.
If you use the IBM logo or the ATT death star in an unauthorized fashion, you get a nastygram from IBM or ATT because you are diluting their brand. Diluting the brand of the Red Cross means that there is an increased likelihood of "mistakes", and a mistake in a combat zone is a bad thing.