Your understanding more-or-less matches what I know about the rules (and there are similar rules in other European countries), although I believe that the threshold is not part of the law anymore. Since 1999, the new criteria are receiving “directives” (Weisungen) from your client and being “integrated in the organization”. Since you are working remotely, presumably don't have a desk, a telephone or fixed presence times at your client's site, this should work in your favor.
The idea is that if you meet those criteria, you are doing what an employee does and you and your employer are using your status to circumvent social insurance payments or employees' rights (protection against unfair dismissal, paid maternity leave, etc.) If you read German, there is a nice overview of the legal background on Wikipedia.
Payments to the pension system are a particular concern in Germany, as the definition I gave above comes from the Sozialgesetzbuch and one organization that's especially active in fighting Scheinselbständigkeit (false self-employment) is the DRV. My understanding is that they regularly send out questionnaires to employers/clients to detect bogus self-employed people and they can demand payment of the pension system contributions retroactively (that would be the main implication and could get very expensive).
Since your employer is in the US, my guess is that this would be less of a concern for you but I have no first-hand experience with this and really no idea whether it's possible to go around these rules or not. Some of the things I read seem to suggest that having your compensation defined per diem, instead of monthly, could help.
Do note that many of the people posting on Toytown would have been working for German employers or at least EU-based employers (hence the need to deal with VAT), which does not apply to you. So here again, it might make your situation a bit easier although I would definitely seek the advice of a tax advisor, at least regarding your income tax situation.
The Handelsvertreter idea you mentioned in your deleted answer might work for all I know but to me it sounds like fraud, plain and simple.