I don't have Android Studio, just the Android SDK. When I launch one of my Android emulators using the Android Virtual Devices (AVD) Manager, I get the following message:
Your emulator is out of date, please update by launching Android Studio.
According to the steps, it seems that one could update the emulator even without Android Studio just by using the Android SDK Manager. But there is no SDK Tools tab.
Is the SDK Manager which is launched as a standalone application different from the SDK Manager available in the Android Studio?
Also, as you can see, everything is up-to-date and the version of Android SDK Tools is even newer than the one that is offered for downloading on the Android website (25.2.3 at the time of writing this question).
So how do I update the emulator? I don't need/want Android Studio.
The standalone GUI SDK Manager is deprecated as of SDK Tools v25.2.3.
The instructions you see in the "Starting Android Emulator" run window are specific to updating via Android Studio. The SDK Manager inside of Android Studio is different than the standalone tool. We added new SDK packages that the old SDK Manager does not understand how to process nor update, including the new Emulator.
You have two ways going forward to update the emulator:
Use the new command line SDK Manager: $ ./sdkmanager --update
Read more here
Use the new SDK Manager in Android Studio
Based on the window output, it also looks like your AVDs might be old, so you may want to create new AVDs or update for the latest features and better emulator performance & stability.