What is the difference between milk (simple) and milk drink?
If the milk is modified by removing the lactose, it is no longer considered milk, and must be sold under a different name, such as milkdrink. The fat content can be modified while still considering it milk however.

The swedish FAQ of Valio (the biggest dairy producer in Finland) is very clear on this. 

http://www.valio.se/halsa/laktosintolerans/fragor-och-svar

I couldn't however find the same quote in the Finnish FAQ (Via google translate).

http://www.valio.fi/usein-kysytyt-kysymykset/

While the legislation could in theory be different between the two countries, I believe this is probably regulated on EU level.