Look carefully. The SQL client does not just "display it in local time", it displays it with a UTC offset. You can be sure whenever you see a UTC offset that UTC is fully recoverable. In this way the TIMESTAMP WITH TIME ZONE field is context independent. It's just UTC.
Conversely, if there is no offset, time zone, or something to distinguish it as UTC (like the Z in ISO8601) then you are just storing "local time", that is the time on the clock in someone's kitchen, somewhere. This is the TIMESTAMP WITHOUT TIME ZONE field and is highly context dependent (in particular, what clock was used?)
Conversely, if there is no offset, time zone, or something to distinguish it as UTC (like the Z in ISO8601) then you are just storing "local time", that is the time on the clock in someone's kitchen, somewhere. This is the TIMESTAMP WITHOUT TIME ZONE field and is highly context dependent (in particular, what clock was used?)