The WHERE clause is needed in order to select WHICH ONE PARTICULAR ROW is to be updated.
`UPDATE ProssessingOrders SET Modified=DATETIME('NOW','localtime');`
would update ALL ProcessingOrders.Modified to the current localtime, rather than just the one row that has been modified. Give it a try.
Of course, the UPDATE in the triggers assumes that ROWID is the unique key for the table. In the absence of a column named ROWID and for a ROWID table, the name ROWID is a magical name that refers to the ROWID.
<https://sqlite.org/rowidtable.html>