the foreign teacher must take the job of teachers' English training with the Chinese English teacher" (not sure what is meant),

I think they mean you'll have to teach classes to the local staff as well as the paying students. You might want to check if the hours spent teaching this class are included in the 20hours (that's quite high by the way) that you will be expected to teach. Private training schools are infamous for the extras that they add on to teachers schedules that can include things like staff classes, English corners, activity classes, marketing events, etc. These are often not included within the contracted hours but are still compulsory.

Personally I'm not a fan of having to do office hours. I can organize my own time and still teach to a high standard. I wouldn't want to be sat in the office twiddling my thumbs just because I 'have to' be there. The problem with china (although this is likely to change in the next couple of decades) is the abundance of cheap labour. It means that businesses don't seem to worry about concepts like efficiency. Chinese staff work long but not particularly smart. Unfortunately such attitudes spill over into the way FTs are managed and if you give them the chance Chinese management will gladly waste your time.