This is how a form widget looks like for a vocabulary with nested terms, when configured to be rendered as a checkbox list:

For D7, you can use the Taxonomy Term Reference Tree Widget module to improve how this widget works, but I believe that all the bell and whistles it offers would be too much for core. We should just get rid of the dashes from the checkbox labels and add indentation to the checkboxes. It would be so much better and more readable like so:

Related: https://drupal.stackexchange.com/questions/85011/how-to-get-rid-of-the-s... https://drupal.stackexchange.com/questions/121053/add-class-to-sub-taxon... https://www.drupal.org/project/drupal/issues/648264 (Core taxonomy.module issue, pending since 2009 - although for dropdowns)
Recent comments
Thanks, the /tmp folder was the problem. I moved it elsewhere and then the updated worked. Thanks again!
Update 1.35.1 appears to install but hasn't
Probably not the reason, but worth to do a quick check: Sometimes the update cache interferes, so if something's odd with updates - some module (or core) claims to have updated, but then...
Update 1.35.1 appears to install but hasn't
Nope, not touched any permissions!
Update 1.35.1 appears to install but hasn't