core_user_get_users_by_field doesn't return any value if the Web Service user is not an admin or the user you are looking for. core_user_get_users neither.

Both of them (user/externallib.php) rely on user_get_user_details_courses (user/lib.php) and this one on user_get_user_details (user/lib.php).

strong*user_get_user_details*strong on line 268 stated the following condition:

if (($isadmin or $currentuser) and in_array('username', $userfields)) {
$userdetails['username'] = $user->username;
}

So emphasis_username_emphasis is not returned.

In strong*get_users_by_field*strong:

if (!empty($userdetails) and !empty($userdetails[$field])) {
$returnedusers[] = $userdetails;
}

As username was not returned before, the second part of the condition is false so $userdetails is not assigned to $returnedusers and the web service return it empty.

Looking for idnumber differs because in strong*user_get_user_details*strong:

if ($isadmin or $currentuser or in_array('idnumber', $showuseridentityfields)) {
if (in_array('idnumber', $userfields) && $user->idnumber) {
$userdetails['idnumber'] = $user->idnumber;
}
}

The or codition is true if we assign to the web service user the viewuseridentity and to the $CFG->showuseridentity the value idnumber.