I found that the "Autocomplete term widget (tagging)" field widget does not save the referenced term, if the name of the term is "0".
This applies to both cases, in which a term with such a name does exist, and to when a term does not yet exist and should be created.
The problem is in the taxonomy_autocomplete_validate() function (line 1639 of taxonomy.module):
<?php
if ($tags = $element['#value']) {
?>
If i change that line to:
<?php
if ($element['#value'] !== null) {
$tags = $element['#value'];
?>
Then it works fine, both for saving existing terms with a name of "0" and for creating new terms with the name of "0".
Not sure if the "!== null" is the best option - but it does solve it.