There appear to be several ways to launch the SDK Manager:
SDK Manager.exe in the root of the Android SDK. 
SDK Manager.exe in sdk\tools\lib of the Android SDK. 
Window -> Android SDK Manager menu in Eclipse 
android.bat in sdk\tools of the Android SDK. 
In my case, it looks like android.bat fails on the line:
for /f %%a in ('%java_exe% -jar lib\archquery.jar') do set swt_path=lib\%%a
As far as what that line is doing... if I manually run: "[path_to_java]java" -jar lib\archquery.jar
It successfully returns: x86_64
But when the batch file runs that same command, I don't know why but it fails with the error message:
Unable to access jarfile lib\archquery.jar
So the variable swt_path gets set to an empty string.  Everything breaks down from there.
The batch file sets the correct value for the variable java_exe.  Others have commonly reported this as a problem, but those workarounds weren't relevant in my case.
People have recommended commenting out the problem line by adding REM to the beginning of it, and adding a line to manually set the swt_path variable, which is a valid workaround:
REM for /f %%a in ('%java_exe% -jar lib\archquery.jar') do set swt_path=lib\%%a
set swt_path=lib\x86
BUT, the critical issue in my case is that it's choosing to load a jar file from either the lib\x86 or the lib\x86_64 folder here.  At some point, things were getting confused between the BAT file error, a 32-bit JDK, and a 64-bit Android SDK.
SO, the workaround in my case was to:
- Uninstall ALL versions of Java
 
- Install the JDK
- You can either use the 32-bit Android SDK and install the 32-bit JDK
 
- Or use the 64-bit Android SDK and install the 64-bit JDK 
 
- But the "bitness" of the JDK should match the Android SDK.  It appears that either of the 32-bit or the 64-bit will work on a 64-bit computer, AS LONG AS the JDK bitness matches the Android SDK bitness.
 
 
Edit "android.bat"
If using the 32-bit Android SDK/JDK, use lib\x86:
REM for /f %%a in ('%java_exe% -jar lib\archquery.jar') do set swt_path=lib\%%a
set swt_path=lib\x86
 
If using the 64-bit Android SDK/JDK, use lib\x86_64:
REM for /f %%a in ('%java_exe% -jar lib\archquery.jar') do set swt_path=lib\%%a
set swt_path=lib\x86_64
 
 
After doing this, I can successfully run the SDK Manager by running android.bat, or from the Eclipse menu (but still not by running either of the SDK Manager.exe files directly).