as Jacque said, you don't need the full Studio, the SDK manager is all that's required. But if you have the studio already, it works, too!
What you need to enter in the LC preferences/ Mobile support is the correct path to the installed SDK, and this can be tricky and confusing. Android creates a bunch of folders all over the system, and many of them hidden ...
For me on Win (with full Studio) it's: "C:\Users\myName\AppData\Local\Android\sdk". On another computer with only the raw SDK it's: "C:\Users\myName\AppData\Local\Android\android-sdk"
In any case it's the one that is shown in the header line of the SDK manager (Your first pict, the selected entry at "Android SDK Location:")
Select the correct SDK folder in the LC preferences, make sure it's the same as shown in the SDK manager, and you should be done. The JDK path should be there then, too. Automagically.
Hope this helped, have fun!
Livecode programming until the cat hits the fan ...