From my experience (I've worked in Thailand, Indonesia, Egypt and the Caribbean:
DAN is actually several different companies under one umbrella - DAN Asia-Pacific, DAN Japan, DAN Europe, and DAN America. As a UK citizen I registered with DAN Europe - but you are covered wherever DAN is present, and furthermore, you can use the local contact centre - so if you are a European working in Thailand you can contact DAN Asia Pacific if you need their services. There is an extra premium for working in North America, but otherwise you are fully covered for liability purposes.
I was not aware that DAN America would not cover liability. This would, however, only apply to US Citizens. As a European it doesn't apply to me - if I were to work there then I would be covered.
I've had to deal with both DAN AP and DAN Europe in emergencies and found their service to be most excellent, although in Thailand and Egypt I also had the benefit of being personally acquainted with a senior representative for the area (but dealt with the company also by phone, which was fine)
PADI require their instructors to be insured in order to maintain active status, and you need to be personally insured in order to do so. Thailand may be an exception, but only regarding your active status, and many employers require you to be insured regardless. In any case, only a very foolish dive professional would not take out personal insurance cover.
(Note: I also have to disagree with the post above suggesting you cannot be prosecuted in Thailand under Thai law. As a foreigner in Thailand you can be prosecuted for anything. I also know for a fact that an instructor working in Thailand can be prosecuted under other countries' laws - my local DAN representative was involved with such a case during my last year in Thailand involving a Japanese diver suing a divemaster after being put on Oxygen with suspected decompression sickness. The diver didn't like it, and the case was brought because the DM (Australian, I think), despite having training in O2 provision, did not have a dedicated O2 provider certification. I actually don't know the outcome of the case as it was still ongoing when I left Thailand, but my understanding is that this in part led to PADI subsequently introducing their O2 provider certification)
All of the centres I've worked for had dive centre insurance, but I was still required to have my own personal cover. It is highly unlikely that any dive centre in a tropical resort would pay for this.
I know Aquamed is popular - they covered a lot of my non-UK European customers (mostly German, Austrian and Swiss) when I was working in Egypt. Most of the instructors I knew in Egypt were covered by DAN due to the presence of the local office. I have no direct experience with Aquamed so can't comment directly, but I have no particular reason to think they did not provide a quality service.
Hope that helps,
Cheers,
C.