Although I almost know what in doing so means, actually, I am wondering whether or not by doing this/that et cetera, could mean the same meaning as the former.
Thanks in advance
Using the word "in" means that the action takes place at the same time as something. However "by" is used to denote the manner or reason something was done.
So "in doing so" actually means "while doing so", but "by doing so" means "because I did so".
I ran out the door, in doing so I slipped on the ice outside.
This means that while I was in the process of running out the door, I slipped.
I ran out the door, by doing so I slipped on the ice outside.
This means that I slipped as a direct result of running out the door - which isn't entirely true. If I had noticed the ice, worn different shoes, or stepped carefully I could have avoided slipping while still "running out the door". The direct cause of my slipping was negligence, not the action described in the sentence.
I suspect many native speakers wouldn't really distinguish in/by [doing so/so doing]1, but to the extent that there is any tendency to use them differently...
I hastily cleared up after the party guests had left, but in doing so I broke an expensive vase
...most people probably wouldn't use by doing so. That's because the in version often simply means while, at the same time, whereas the by version normally implies as a direct, inherent, and/or intended consequence.
1 As can be inferred from the lack of a clear consensus in responses to (presumably, a different) nima's question on this dictionary forum last year.
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