I have a 6-year-old iMAC desktop and I assume it has Adobe Flash Player on it. I've been prompted over the last year to update Flash Player and I've also encountered websites that will not play videos and instead simply shows the message "Flash out-of-date." On these multiple occasions, I have followed all the prompts to update Flash Player but nothing updates or changes. I have just created a new Adobe id because our old one is terribly old, uses an email we don't use anymore, and, when I tried to use it on this forum today, it said the password was incorrect.
In your case, I recommend running the Flash Player uninstaller to completely remove Flash Player that's installed on the system. Download the uninstaller posted at Uninstall Flash Player for Mac OS . Don't worry about deleting any files, etc. Just download the uninstaller file, completely exit out of all browsers and apps that may use Flash Player and then run the uninstaller.
After downloading the uninstaller, launch your browser and navigate to https://get.adobe.com/flashplayer to download the installer. Note that this page utilizes browser/os detection to offer the appropriate version of Flash Player for the browser and OS. For Safari and Firefox this is the NPAPI plugin. For Opera, and other Chromium-based browsers this is the PPAPI plugin. Note that Google embeds Flash Player in Chrome and there is no need to install Flash Player separately.
Note that essentially all browsers block Flash Player by default. After installing Flash, you'll most likely need to enable Flash Player, sometimes, on a per-site basis. The Flash Player Help tech note contains links to enable Flash Player in the various browsers.
Be aware that the web is full of sites which tell you urgently to update Flash, and have very plausible pages with the Adobe logo and all ... and are there only to infect your computer with really bad stuff. NEVER follow update messages. Go to Adobe to check if there is a new one.
Better yet, either use a browser that automatically keeps Flash Player updated (Google Chrome, or IE and Edge on Windows 8 and higher), or enable automatic updates for Flash Player on the system. You can then browse confidently, knowing that any update notifications for Flash Player are bogus, and can be safely ignored.
To enable automatic updates for Flash Player:
Go to Settings/Control Panel
Choose Flash Player
Click the Updates tab
Choose Allow Adobe to install updates (recommended)
If you're ever wondering if you have the latest Flash Player, you can also just go to this tab and click "Check Now". If an update is available, you can launch the update there.