Should the time zone be recorded in case of timestamp or not?

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WITH temporal_column AS (SELECT table_schema, table_name, column_name, data_type, ordinal_position
FROM INFORMATION_SCHEMA.columns
WHERE data_type IN ('timestamp without time zone', 'timestamp with time zone')
AND (table_schema, table_name) IN (SELECT table_schema, table_name
FROM INFORMATION_SCHEMA.tables WHERE table_type='BASE TABLE') AND 
table_schema NOT IN (SELECT schema_name
FROM INFORMATION_SCHEMA.schemata
WHERE schema_name<>'public' AND
schema_owner='postgres' AND schema_name IS NOT NULL))
SELECT table_schema, table_name, column_name, data_type
FROM Temporal_column
WHERE EXISTS (SELECT 1 FROM Temporal_column WHERE data_type='timestamp without time zone')
AND EXISTS (SELECT 1 FROM Temporal_column WHERE data_type='timestamp with time zone')
ORDER BY data_type, table_schema, table_name, ordinal_position;