While reading a book, I saw a phrase which distraced me. ("on entering the door") And I searched the internet for more information on this usage "upon/on doing". Then I found a very competent comment here

As far as I understand, "upon/on doing" make a connection to the main clause but the temporal relationship is not obvious. It could refer to a time before the main clause or after the main clause or even simultaneous with the main clause . What I want to ask you is that:

What is the different between "on doing"" and "in doing" if both "on doing" and "in doing" refers to some point in the process of doing something? Can I rewrite the sentence as I have done in the sentence below?

1- On entering the castle, he is led to a banquet hall where he meets the Fisher King and sits down with him for a feast.

1a- In entering the castle, he is led to a banquet hall where he meets the Fisher King and sits down with him for a feast.

2- On purchasing the phone, I was assured that this was the latest and the best model available there.

2a- In purchasing the phone, I was assured that this was the latest and the best model available there.

or vice versa:

3- In running out the door, I slipped on the ice outside.

3a- On running out the door, I slipped on the ice outside.

4- In studying with her, I realized she was deeply interested in programming.

4a- On studying with her, I realized she was deeply interested in programming.