When editing a template there are three ways in which those edits can be saved, and the UI should provide an affordance for users to select their preference. To remove any doubt or confusion, this should ideally be possible before engaging the multi-entity-saving flow.

The simplest and most obvious method is to simply update the current template and overwrite the previous version. Just like when you edit a post or page. This would likely be the default selection in most cases.

The second method is to create a new template with greater specificity. That is to say, if I am editing the page template, it should be possible for me to create a new page-$id template.

The third method is to create a new generic template that can be applied to any post or page. The equivalent in traditional theme building would be a $custom.php template.