now() and statement_timestamp()

According to the documentation. now() should yield the timestamp of the
current transaction, while statement_timestamp() should yield the
timestamp of the current statement.

In case of Autocommit, both should be identical, in case of a running
transaction, they should be different.

Funny enough, psql and executing the same statement from within an
application yield different results. I tried a Tcl script and DBeaver.

Statement: |SELECT now(), statement_timestamp();|

psql: |2021-05-07 13:31:10.135851+02 | 2021-05-07 13:31:10.135851+02|

DBeaver: |2021-05-07 13:31:00.633784+02|2021-05-07 13:31:00.633913+02|

Tcl script: |{now {2021-05-07 13:24:54.659837+02} statement_timestamp
{2021-05-07 13:24:54.65985+02}}|

So in psql now() and statement_timestamp() are identical when not in a
transaction, while in the others the two values vary a little bit,
roughly 100 microseconds.

How come the same statements aren't behaving identical. I'm using a
current PostgreSQL server on a Linux machine.

--
Holger Jakobs, Bergisch Gladbach, Tel. +49-178-9759012