Actually I did notice that post on this forum on the unconnected AVCC, so in the custom board I referred to I connected AVCC to VCC and AGND to GND on the PCB itself. However I have this problem on that board too and there the nonzero value is more than '200' (a small error from VCC, as opposed to a small error from zero). However I reckoned I could not ask for support on a custom board so I thought I'd try this on the Digispark Pro and that's how I reported that issue. My feeling is that all other things being equal, the mere usage of Serial should not change an ADC read. So if AVCC being unconnected could cause erratic reads, that erratic read should happen whether you have a totally unrelated call to Serial or not as in the commented line. Besides that nonzero read is a constant 0 and does not happen with reading A8 - if it was a bad reference it should happen with A8 as well as A11!
Anyhow, I appreciate your efforts in helping me try to resolve the issue. I'll post a resolution if I find one. Thanks Ralf!
-Ram