Difference between "on doing" / "in doing"

Question

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.

Answer 1

"In doing" generally means "by means of", and is similar to "on doing", but without a direct temporal relationship.

Example: In reading his textbook, Harry learned the basic principles of mathematics.

Equivalent phrasing: Because he read his textbook, Harry learned the basic principles of mathematics.

Reading his textbook has been a continuous process, and the result of learning mathematics is not necessarily a result that comes immediately after completing the action of reading the book.

As Colin Fine mentioned in a comment, "on doing" means "immediately after" and sometimes implies that the following result is a consequence of the action.

Example with causal link: On finishing his lunch, Harry realized he had taken an extra 15 minutes over his half-hour break.

Equivalent phrasing: When he finished his lunch, Harry realized he had taken an extra 15 minutes over his half-hour break [implied: because he had taken so long to eat].

Example without causal link: On finishing his breakfast, Harry realized he had forgotten to cook enough food for the family.

Out of the four examples you gave, the first three are correctly expressed with the phrase "on doing". Each of the first three could be reworded as "When something happened, another thing occurred." The fourth can be written either way, albeit with two slightly different meanings. If written using "In studying ...", the sentence implies that the realization that the study partner was interested in programming was a direct result of studying with her. That is not necessarily the case with the phrase "On studying ...", but it is an open possibility in that situation.