I'd like to get the install log files from your Mac to investigate why the online/web installer doesn't work. Please read through the Where do I find the Flash Player installation log on the Macintosh? FAQ. Enable verbose logging and configure your to obtain all 4 log files for the same installation attempt. Run the installer until it fails, or returns an error. If an error is returned, take a screenshot of the error. After taking the screenshot (of applicable) close/exit out of the installer. Gather the 4 install log files and upload them to cloud.acrobat.com/send using the instructions at How to share a file using Adobe Document Cloud . Post the link to the uploaded files, and the screenshot (if applicable) in your reply.
Alternatively, the offline installer is posted at Installation problems | Flash Player | Mac in the 'Still having problems' section.
Please do provide the log files so we can attempt to determine why it's failing on you system.
I've been having this same problem for the last year or so. Every single update on my MacBook Pro, for the last couple or so OSX versions, has failed via the traditional update mechanism, and I end up searching for the offline installer link. The offline installer always works. I'd be happy to send y'all my log, but I suspect this is not an uncommon issue. It's a very common experience for most of my friends who also use Macs.
Thank you for including the offline link in your reply. Once again, using that allowed me to install the latest update after having the downloaded installer via the Flash update mechanism fail.
What OS X version do you have? If you provide the log files we can view them and see if they provide information as to why the installer is failing.