Page title module fallback problem
You need to choose a specific token context from those available such as [current-page-what?]:

[current-page-number]   The page number of the current page when viewing     paged lists.
[current-page-path] The URL alias of the current page.
[current-page-path-raw] The URL alias of the current page. Warning: Token value contains raw user input.
[current-page-title]    The title of the current page.
[current-page-url]  The URL of the current page.


You should specify your fallback global page titles under '/admin/settings/page-title' and for specific node types, such as 'story' 'page' or 'blog' for example. 

You should input the page title tokens as follows;

Example 1:  [current-page-path] [page-title]
Example 2:  [page-title] | [site-name]
Example 3:  [page-title] | [site-slogan] - mysite.com


Note that you can insert symbols or text between the tokens or after.  In this case we've inserted the vertical bar '|' and a hyphen '-' plus the site name 'mysite.com' within the pattern.

I think you've understood correctly.

Although the example above relates to Drupal6, I've tried to recreate your problem in Drupal7 and can't because the default [current-page:page-title] | [site:name] always kicks in, as it should.

Have you tried completely removing the Page-Title module, running Cron, updating the database and installing the module again?