Those are coconuts they are tapping in the video.
Toddy palm is Borassus fabelliferus. The borassus is tapped in the same way as coconut to make sugar and also to make toddy wine, and toody spirits (high grade alcohol). The same can be done with coconut. This is common practice in India, Malaysia, Indonesia,Thailand, Myanmar, etc.
Yes the plants you saw in Florida are probably Borassus aethiopium. The trees i saw in Fairchild gardens were Borassus aethiopium. You can probably make smae products from it. But the fruits from aethiopium are smaller and not as good as fabelliferus.
You can see lots of info in the forum previously posted about this and photos by doing a forum search.