How to ensure that two or more simultaneous requests to a MySQL table don't interfere
Transactions are the way to go.

... I found out that while a row is being processed by a transaction, other transactions can still come along and modify that row ...

Cite your source because, frankly, it's just plain wrong.
The whole point of Transactions is that they give you exactly the kind of &quot;atomicity&quot; of change that you're looking for.
Table locking is far too broad a brush to use here.
Locking many rows just update one is not efficient, creates more contention, more waiting and just slows everything down.  To draw a parallel, it would be like trying to deliver a letter to one house by blocking up every letterbox on every other house in the city.