The error was that jobCheckTime was changed everytime, even if job was not changed at all.
This lead to message "job saved" each time you saved an item with jobs, even with no change. So you never had "no change" message.

I see, indeed, then we should update checkTime if $old->value != $new->value, right?

No need.
It is already done in saveJioblist.php

    if (isset($_REQUEST[$checkName])) {
        $checkedCpt++;
        if (!$valLine->value) {
            $checkedCptChanged++;
            $valLine->checkTime = date('Y-m-d H:i:s');
        }
        $valLine->value = 1;