How do I get records proving that I was an Erasmus exchange student?
I ended up contacting the Academic Services / Student Requests Information Team by email. 

They said immediately they would provide a document, and they delivered; Pretty much professional and very attentive people. 

They found my records from 20 years ago, but not my grades. Fortunately my dissertation grade is not so important, as I have it in another set of documents.

I already have the document in electronic form, and they also sent a paper copy.