[Eliot Ness]

4.  Unless the tube chart indicates otherwise, it is normal for the neon lamp to glow when only one of the slide switches is pressed to "P" position. This simply indicates filament or heater continuity.

It is not worded very clearly, but what it means is that (unless noted otherwise on the chart) ONE of the switches should show a short because it is the other filament pin.  In the case of the 12BE6 pin 3 and 4 are the filaments:

http://frank.pocnet.net/sheets/093/6/6BE6.pdf

So since pin 3 is already selected as the filament, pin 4 will indicate a short since it is the other filament pin. 

The dark area you see is probably the tube getter:

https://www.tube-town.net/info/doc/dark-getter.html

Finally, if your tester is from the mid-60's it is going on 50+ years old.  That means any capacitors in your tester should probably be changed.