I've tried the expression on step 2 but that didn't work.

How can i create the logging asked in step 1?

In step 2 you talk about an alternative method but i don't know how to accomplish writing another value twice.

Let's separate this into two possible problems: 

1: Your trigger does not get activated. (Try creating a logging record at the beginning of the python code expression to verify if it activates)

2: Your value update itself does not work. Try to see if you can have the python code work using "record.write({'write_date': datetime.now()})"
Alternatively, while it is suboptimal, you can also opt to write another value twice (once to a new value and then back to the original). This will have the write_date updated for sure.