As much as thats annoying, it does make a lot of sense. If you can assert that everything which bears that logo is going to be medically related (and not just something random), especially in wars and the like, means you can be more certain of someones intentions.
Both the cross of the Knights Templar, and the similar Maltese cross used by the Knights Hospitalier are quite distinct from the straight cross the Red Cross uses. I very much doubt anyone anywhere where variations like that are likely to be seen would confuse them.
It makes sense, but the red cross is way too generic a symbol. And it makes perfect sense for a health pick up - it's a +, meaning it adds and it's red for health. And I don't see how anyone would mistake a health pickup in a game for a Red Cross™ worker.
We have a product/website that used a blue cross in the logo. Blue Cross Blue Shield started hounding us about it, so we changed the logo to a green cross.
