How can I translate "非语言表达方式"?
Your understanding is right. 

"非语言表达方式" means expressing things without using language, here the word "language" means just spoken and written language, gestures are not considered as "language" here although they can be refereed to as "肢体语言". 

The reason is that you need to understand the word "language" here by context. The word "language" from the first half of the sentence means spoken and written language, e.g. Chinese here, so it still means that in the second half of the sentence.

Actually, the reason we Chinese sometimes call gestures "肢体语言" is that it serves the purpose of expressing thoughts and emotions, just like what we can do with the "real language". 

There are other similar phrases like "艺术语言", which means the way of expressing by practice of art.

Hope this helps :)