There are actually several different types of rashes that Lyme produces, but if a person gets a bulls eye rash, it's definitely from one specific strain of Lyme - Borrelia burgdorferi, and possibly from Borrelia mayonii. That's all the tests are designed to look for as well, which is yet another reason the testing is so faulty.
According to ILADS: "Fewer than 50% of patients with Lyme disease recall a tick bite. In some studies this number is as low as 15% in culture-proven infection with the Lyme spirochete." www.ilads.org/lyme/about-lyme.php#sthash.3n5vZQXd.dpuf
I've never had any known rash, unless it was in my hair - but I doubt that, simply because as a child who loved being outdoors, I had a tick check daily and any rashes would have been a cause for concern.