Thanks for the question, Vectra. It reminded me that correction can be so difficult, and occasionally subjective.. Below I have added my comments in bold to yours on the first paragraph only.
The meeting was on Saturday afternoon, so she decided to leave early that day. In order to be on time she took a taxi to the station. However, it was an unnecessary precaution, since the fast train to Reading was delayed (I think - had been delayed) I understand why you think so, but I think that I would accept the student's version also. due to floods. Upon the X arrival in Reading, she took a coach to the Heathrow Airport (the is not used in front of airports' names correct), but it was again (I would prefer 'it too' or 'that too'. The coach wasn't delayed 'again' - it was the first time. But this is rather pedantic.). delayed (I think - it was delayed too agreed,)– this time because of road works. She happened to arrive at the airport just before the (?)takeoff and stormed through the terminal towards the security check point at once, since she only had one handbag on her and no baggage (as we study British English, luggage is more suitable, I think yes, but 'baggage' is overtaking 'luggage', I feel.). After the check she ran fast to her gate (her is wrong here. It is just the influence of the Russain language. The is better I am happy with either; in fact, I prefer 'her'. I hope she had a comma before the non-defining relative,) which happened to be on the other side of the airport, and then back again to get her only bag which she happened to leave (I think - happened to have left She had happened to leave?) on the security desk.
This also made me think of the problems non-native speakers have when dealing with advanced students. You are right - the student's English is excellent, and your correction is perceptive. It's just that it is so difficult these days to decide what is fully acceptable, marginally acceptable, etc. As a reasonably well-educated native speaker with many years of teaching experience, I have problems; for example, I feel intuitively that the past perfect is used less by most speakers today than it is by me or the course books, and that this is acceptable. If they do have a fault, teachers who are non-native tend to be a little harsher that natives (as with the 'her gate' above).
It will be interesting to see if others come to this post and correct my corrections