This is exactly what you should see in the automation lane's edit filter menu when you've created an automation instance mapped to the drum's volume slider in Battery 4. If you have a synth track that's connected to Battery 4 and you don't see Battery 4 in the automation menu, then something is seriously amiss. 
 

 
 

I have a dozen or so MIDI tracks.  Their output is Battery 4.
 
I also have one non-MIDI track for the drums as well; its input is Battery 4.  This is the track to which I apply audio effects, volume changes (usually just a fade out at the end), etc.  It's the one I tried to use to automate the volume slider on the ride.
 
I'm not sure if the "audio" track is actually a synth track or an audio track.  Is there a difference between the two?  How can I tell what type of track it is?
 
 

An audio track has the icon that represents a stereo waveform. A synth track has the icon that's a combination of a MIDI plug and a keyboard. Synth tracks, essentially, are audio tracks whose input is the output of a synth. 
 
Apart from MIDI tracks, the other type of track you're likely to run into is a Simple Instrument Track, which combines audio and MIDI. It's icon is just a keyboard. 
 
Open up an automation lane on your Battery synth track and click the edit filter that says "volume." There should be an entry for Battery 4 there. If possible, post a screenshot the same as mine and show us what it looks like if Battery 4 isn't there.