Legal issues from a radical community angle

A trademark is used to refer to a specific product; a service mark is used to refer to a specific service. Typically, such marks may be fairly and freely used as-is and without change to refer only to the actual organization, product or service with no other association inferable or implied. Full stop. Using a trademark such that it also represents something else or implies a non-existent relation without license to do so is wrong. And illegal in most places. As I understand, this is basically the policy Debian adopted.

For your project, you would have to design your own trademark that refers solely to your project, implies no association with any other organization (via others' 'adapted' trademarks) and does not lead anyone to infer such association. In your documentation, it would likely be acceptable to to say something like, "This project is based, in part, on the freely available Debian® GNU/Linux software system" and display the logo. Here, I believe it would be fair to use the trademark image because it refers solely to Debian and implies no association between the two projects.

It's much the same as saying, "I only use Gildan® t-shirts when I print wearable billboards", "I used my Pentax® camera to take these pictures" or "I started with a Dodge® pickup truck with a Cummins® engine and built this 10-second drag racer." Here, the trademarks refer only to the original products and/or manufacturers (t-shirts, a camera, a pickup truck and an engine) and imply no other relationship between me and the trademark owners.

No businesswoman interested in ensuring that her trademark refers solely to her product(s) would allow anyone to incorporate her trademark into another without explicitly granting license to do so and without setting up legal indemnifications protecting her and her interests. Debian is no exception.