When saving a new template, the only folders available as destinations for saving the template are those already populated with template files. If you delete a template file from a target folder then restart the program, this folder will no longer be available to save templates in.
*** Bug 61258 has been marked as a duplicate of this bug. ***
This is not a bug. You just need to set the template location in:
Tools -> Options -> Paths -> Templates
If you set the path then "My Templates" links to the folder and works fine.
Marking as NOTABUG. If this doesn't solve your problem please reopen and I'll investigate further.
Thanks for reporting!
The template path is set by default upon installing LO. Having the template path set does not cause a "My Templates" folder to appear when saving templates. It is available as a destination if you want to move a template from one folder to another, but it is not available as a folder for saving into.
The only reason, in my installation, that I can save templates at all is that I also installed the MediaWiki extension, which happens to add a MediaWiki template folder. I can save into that folder, and after I've saved the template I can then go back into templates through File->New (Template) and copy that template from the MediaWiki folder to "My Templates". Anyone who has installed without installing the MediaWiki extension will not be able to save templates at all.
related to bug 60950
*** This bug has been marked as a duplicate of bug 60593 ***