Could the red and white circles (bulls eye) around a tick bite mean anything other than lyme disease?

Depends on location : In the southeastern usa, a similar rash can occur following the bite of the lone star tick called southern tick associated tick illness or stari. Progression to arthritis or neurological disease appears not to occur.

See below: Usually, this is the cardinal sign of lyme disease but theoretically, other diseases like ringworm could be coincidentally present.