localtimestamp or curent_timestamp for default value
You write "I don't need the time zone information", however datatype TIMESTAMP WITH LOCAL TIME ZONE has time zone information stored. 

For a TIMESTAMP WITH LOCAL TIME ZONE column I don't see any difference between CURRENT_TIMESTAMP and LOCALTIMESTAMP either of them should work in the same way.

CURRENT_TIMESTAMP returns current data and time as of datatype TIMESTAMP WITH TIME ZONE, whereas LOCALTIMESTAMP returns a LOCALTIMESTAMP value. Both of them have to be implicitly converted to a TIMESTAMP WITH LOCAL TIME ZONE datatype.