Why does Lamport clocks increment on both message received and sent?
From Leslie Lamport Time, Clocks, and the Ordering of Events in a Distributed System:

The space-time diagram of Figure 1 might then yield the picture in Figure 2. Condition C 1 means that there must be a tick line between any two events on a process line, and condition C2 means that every message line must cross
a tick line.

I think my confusion was on what actually is an event.
In Lamport's paper and in Wikipedia, sending and receiving the messages are the events. In that case, incrementing the counter both when sending and receiving a message make sense, to ensure that a clock tick separate those events.
In my case, the data structure modification are the events. The message passing between the processes is irrelevant. In that case, I believe that incrementing the counter only when the data change make sense.
Now, if applying this Lamport clocks principle this way still qualify as Lamport clocks and holds all the related properties, I do not know.