The best way to figure it out is removing the older version and downloading the latest one. Then you log in and all the synced things come again

This has happened before and suggests a modification to the Firefox version file.

I am looking for the method the new installation of Firefox verifies so I can determine why the old version is consider newer than the latest version of Firefox.

Your system details shows; User Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.10; rv:45.0) Gecko/20100101 Firefox/45.0

It is possible that you have installed Firefox 44.0.2 after Firefox 45 got released officially and that is confusing the Mac OS because of the file time stamps.

If you can't update via "Firefox > About Firefox" then you run the latest version.

See:

• https://support.mozilla.org/kb/Finding+your+Firefox+version

No. Make no mistake that I have not installed any other version before installing the new latest version 45. As I have already stated this has occurred before.

I had version 44.02 tryed to intll verion 45 nd received a message that a new version already existed.

Does anyone know what file is checked to determine the file version?

Sometimes, the updates are sent out by sectors. Try again in a few days.

There is now a Fx45.0.1 out.

Does anyone know the check process for Firefox when upgrading so I can determine if my current version 44.02 has been changed, hence why it detected a newer version when I installed version 45?

Firefox will sometimes start an upgrade and not suceed so sometimes that needs to be cleared manually, but it usually works pretty well. To both check the version, and trigger an update if one is available use menu > help > about Firefox See Which version of Firefox am I using? If you are on Fx 45.0.0 You should upgrade to 45.0.1

Edit Sorry wrong link it should have been Find what version of Firefox you are using

This is a security issue. Firefox 45.0 is not reporting any updates. Yet under "trouble shooting information" when viewing update information it shows no new updates. I have not performed an update and it still shows version 45. and not 45.01 which it did yesterday March-24-2016

why?

Did that sort itself out after a restart ?

If not try clearing any pending updates as explained in this old article

• Updates reported when running newest version_delete-update-configuration-files

   Delete the following files: 

   active-update.xml
   updates.xml
   The updates folder and its contents