As I just come from uni I asked my linguistic prof again and he said that protestant and evangelic are both right to translate evangelisch.
To preach the gospel is evangelize.
And my Langenscheidt dictionary says exactly the same.

I only can relate to what I've learned and what teacher/profs told me and what my lovely Langenscheidt says.

I always thought evangelical was extreme protestant, like the complete opposite of Roman Catholic. But you can also be evangelical catholic, I don't really get it.
No, that's not possible. Either you are evangelic OR catholic. Both religions belong to Christianity. The difference is small though (e.g. the evangelics don't believe in Mary's Immaculate Conception while the catholics do).
I'll try to explain what I know or what I've learned what the differnce is:
Christianity is the generic term.
Within the Christianity you have either catholics OR evangelists.
The word protestant was formed from evangelists.
Evangelist is the self-designation that came from the reformation of Luther and Calvin and the emerge of their churches and Christians.
The basis of their religion is the gospel about the message from God's grace.
Protestant is a very wide term for cultural and spiritual phenomenons who emancipated themselves from the generic term. Furthermore protestant is more a political term than a religious one because the protestant church developed from the protest in Speyer 1529 about a catholic resolution.
Today is the day of my worst English. Congratulations.