When a Flash Player version becomes outdated, Safari or Firefox will pop a small dialog to let you know. Since a well-known scam site does this too, you should not click the Safari dialog, but just go directly to the official Adobe Flash site.
No browser, or System Preferences can be running during the Flash Player installation. When the installation is complete, you will be directed to the general Adobe site. You are done.
In Safari Preferences : Security : Internet plug-ins: Check Allow Plug-ins, and then click Plug-in Settings… √ Adobe Flash Player and When visiting other websites, select Allow. Additionally, Flash Player may offer a Currently Open Websites panel (e.g. here in the discussions.apple.com site), which I have checked to Allow. Here is what mine looks like:
Once you click Done, quit and relaunch Safari. You should have an operational Adobe Flash Player now.