Let's start with the definition of the word "chronic".
According to Miriam-Webster:
adjective chron·ic \ˈkrä-nik\
medical : continuing or occurring again and again for a long time
: happening or existing frequently or most of the time
: always or often doing something specified
Full Definition of CHRONIC
a : marked by long duration or frequent recurrence : not acute <chronic indigestion> <chronic experiments>
b : suffering from a chronic disease <the special needs of chronic patients>
a : always present or encountered; especially : constantly vexing, weakening, or troubling <chronic petty warfare>
b : being such habitually <a chronic grumbler>
This is what "chronic" means in English. Perhaps it has a different meaning elsewhere?
To me, it is obvious that an ongoing infection that lives on even after trying to kill it with antibiotics should be considered a "chronic" infection.
However, damage due to a past infection can be considered chronic pain or chronic disability. As an example, Polio was a devastating disease that left many people permanently disabled, but a lesser known issue is that it left some people with life-long chronic pain as well.
I wouldn't consider those people as having a chronic infection, but I would consider them as suffering from a chronic medical issue.
Herpes is a chronic infection that cannot be completely eradicated, although it can be controlled somewhat. Herpes is a chronic disease - it never dies completely.
Lyme disease may or may not be a chronic infection for some. I think we may not actually know yet whether Lyme disease is a chronic infection or not. What we do know is that it has left some of us with permanent health issues and others with chronic, recurring health issues.
Call me crazy, but it seems chronic Lyme disease is when you are infected despite antibiotic treatment and experience periodic resurgences of symptoms.
If the infection is dead, but you are still experiencing pain, that would not be a chronic Lyme infection. I suppose, if your health problems originally stemmed from a past Lyme infection, some people may call that chronic Lyme, but that should probably be called something else. I know! How about "Post-Treatment Lyme Disease Syndrome or PTLDS?
Hmmm...A little clunky. Doesn't quite roll off the tongue.
You know what's easier to say? That's right: "Chronic Lyme". If doctors didn't have their panties in such a tight twist about the fact that Lyme can survive some antibiotic treatments, then we would all probably be calling it simply, chronic Lyme.
Even still, I'm pretty sure when people say "chronic Lyme" they're talking about an infection that won't die.