It is a bit confusing at first but you have it correct.
Adwords is the Google program for advertisers where you create your ads and bid on certain keywords. The Ads appear on their search network which includes Google, Ask, AOL, Earthlink and some others.
You can optionally bid to have ads appear on their content network of websites which include sites like NY Times, Howstuffworks, About, a whole bunch of sites.
Adsense is for site owners and free to join, you display the (Adwords) ads on your site via code Google provides and earn a certain amount when someone clicks on the ad. The amount varies depending on the keyword.
With my (non-gambling) e-store they’ll be a bunch of sales from NY, then another time a bunch from Hawaii, etc.
Last year, when I was trying to advertise the slots ebook, Google told me no gambling sites as well. When I said that I see Adsense ads for gambling sites, they replied that those sites are not in the US.