I think the key is the meaning of the word "distribute." I believe it is generally accepted that using MySQL in a web based application provided as a service doesn't constitute distribution (though there are some questions about situations when the app is hosted on servers operated by a 3rd party).

I was suspicious, but I read MySQL's page on licensing and that seems to be true. I'm quite shocked about it, I thought you were free to use MySQL in non-open products without having to buy their commercial license.

you can use it all you want. Just don't distribute it. e.g you are prohibited to take it and make tweaks and release your own DB software...