Thank you for your advice. There are some things I will certainly try.
Your assumptions are right. I'm the one that designs the research, does the analysis and writes the paper. When I ask for advice during the data-preparation or analysis phase, both my professor as my second promotor are very helpful. However, when I'm in the writing phase, it seems I'm only slowed down by my second promotor. My professor says he doesn't have time to check my manuscripts, so he mainly comments on the tone and content of the manuscript, which is great. However he doesn't want to get involved in any differences we have when it comes to writing.
My fiance is a native english speaker and sometimes I check some text with him. He agrees with me on the grammatical errors my second promotor made, although I'm still also still learning a lot from him. When I'm writing I always have a dictionary and thesaurus close by and I always check every word or sentence I'm unsure about. Although my sentences are not always as smooth as those of a native speaker, I'm pretty sure they are at least correct.
Basically the problem breaks down to two smaller problems. The first is style, the second is grammar.
I spend a lot of time making sure that my manuscript is easy to read, doesn't repeat the same words over and over again, and keeps the reader interested. My co-author has a different writing style and tries to force this onto me by for instance replacing words with their synonyms. She also swears by this fixed structure of the text, which I have never seen in any article or writing course. She tries molding my manuscript into this structure, she herself uses for all her articles. I guess you can say that this is a matter of taste. I prefer texts in which arguments are ordered so they are easy to read, she prefers a fixed structure. I prefer some variation in my words, while she likes to use the same words over and over again to keep things simple.
The second issue is the grammar thing. Here, she is mostly wrong. For intance she often makes mistakes on the verbs. When I write "Many European countries are faced with..", she corrected this with "Many European countries are being facing".
I guess I could indeed solve the grammar issue, by explaining why I don't use her suggestions, until she gets the point. However, the style issue is something completely different. On the one hand, the articles will be in my thesis, so I feel they should be written in a way that I prefer. Of course I look at other articles to discern the common style of writing in the field, so it's never exactly in a way that I would prefer. On the other hand, I would like to keep my co-authors satisfied. I guess we will both have to compromise on style...