I tried to find an explanation for the apparently strange statement of your teacher, I hope you get what I mean:
Am Anfang hat Judith schlecht Spanisch gesprochen.
Could be translated as: At the beginning (or at first) Judith had spoken in a poor/bad Spanish
This statement does not necessary means she had Problems with the language itself, for example maybe she could understand it well, but had very basic vocabulary.
But if one could deduce that she had pronunciation problems … syntax problems … I’d rather agree with you and say she had problems with that language.
Well as we know, when we learn a new language at the beginning most of us keep thinking in our own, so as a German I would build for example Spanish sentences according a German sentence structure and so on. It’s really hard to say that these problems have nothing to do with the language. But maybe somebody who is learning a language and having such “problems” would say “no I have no problems”, as this is usual at a beginner status!
Judith hatte keine Probleme mit der Sprache.
This simply means: Judith didn’t have any problems with the language!
As you see the margin allowing us to mark this statement as richtig according to the first one is very thin but it could to some extent be accepted. While marking it as falsch seems clearly favorable! Espacially as anything that would come in mind when reading the first statement to explain why the second could be correct could be objected because of the word keine/any!