That's exactly it. GDPR doesn't particularly care where you're storing the data, nor whether you have a "European presence". If you're storing data on people from the EU, from the EU's perspective, you're required to follow GDPR.

From an enforcement point of view... if you have no EU presence, it'll probably be pretty hard for them to actually collect on the fines they levy, but as a director, travelling to Europe could be problematic. It's not entirely clear here, but it looks like there's a possibility of directors being held personally liable.