Let's explain the basics here.
Ok, now,
I see why the first design tends to perform better. I'd still check it. Now believe me even if you feed the primary with AC+DC, the only current that you can get on the secondary will be AC, regardless if you drive the core to saturation or not. Of course if you do you'll have some losses there.
Let's see.
As primary gets series of pulses, the core's flux goes from 0 to maximum flux, (not - to + max-flux as if you'd power it with AC, drawback: you can use only half the transfer energy capacity of the core but that's not the issue here) As the flux rises you'll get the EMF in one direction, as it falls to zero, you'll get the emf in other direction. NO DC COMPOUND!
The most probable reasons for blackening one side of the tube (thus emmiting mass of one heater/electrode) would be:
-unbalanced duty factor, The duration of pulse should be as much as just to drive the core to saturation or even better, just not to. Duration of off period should be the same (as to allow the core to recover to zero). This can take some fiddly tuning and adjusting.
-the second reason would be very different rise and fall times of those pulses hence one electrode would handle higher current, the shorter time and vice versa. That can pretty much evaporate the electrode during time.
Otherwise, it should just work fine.
The first design makes beter use of transformer, hence the smaller is needed, but not really a standard one, and is easier to tune. The second uses one transistor, and a bigger transformer, but can be one off a shelf, and is a bit more fiddly to tune. Choose for yourself.
Personally, I would get a Chinese one for almost nothing and probably wouldn't bother too much if not going for a production of it.