Basically what it says on the tin. Having read though some of the materials on the issue, I am baffled by how recklessly the word is used, given the consequences of such usage.
Pedophiles are the people with sexual attraction to prepubescent children. It doesn't matter whether they do or don't act on that attraction; in fact, many don't. It is a sexual interest/mental condition that cannot be reliably changed.
Child molesters, on the other hand, are not necessarily pedophiles - in fact, 50 to 75% of child molesters do not have pedophilic interest.
Both facts can be sourced from the respective Wikipedia article and more info can be found in respective research.
Why does this matter?
Because the current use of the word reinforces stigma around pedophilia and makes it less likely for people with pedophilic disorder to reach out for help for the fear they would be outed and treated the same as actual child abusers.
This, in turn, makes those in a vulnerable position more likely to cross the line and get into the category of child abusers instead of coming for help. Also, it heavily affects people who did nothing to deserve such treatment.
What should we do?
We should leave the word "pedophile" to the context in which it belongs, which is the mental health and sexuality spheres, and avoid using the term to describe sexual offenders against minors. At the very least, one would most likely be wrong. At most, one would participate in the cycle of child abuse.
I think you contradict yourself here. You say there should be a stigma about having these thoughts but then say people should get help and not be stigmatized for it.
If there is a stigmatization about it, then that is going to keep people from bringing it up and getting help due to fear of being attacked either physically, financially, or emotionally.