Higher morale indeed

[garbare83686]

The question of morale has come up in our little battles, to wit: "All units of attacking side have higher morale than defending units." My understanding of "morale" is the level of morale in the unit (i.e. Is it Disrupted/Demoralized or no.)

No. In this case it is the morale value of the units involved, leader and/or terrain morale modifiers are not considered. So, if all the units on a side have a morale of 8 are fighting a side with units of a morale of 7 (or less), the side units with a morale of 8 would get a +1.
 

I have an opponent who seems to take it as the SIDE with the higher initiative gets the +1.

No.
 

I have been playing it as ALL units must be demoralized on the defending side to qualify for the +1. (Perhaps "Disrupted" units should be counted also?)  (Morale being morale and not initiative.) 

That is a separate modifier from the one in the first question. If all units of a side are demoralized, the other side would get a plus +1. Note that this would always be given to the attacking side as demoralized units may only defend or attempt to rally. For example: If both sides have a morale of 8 for their units, side A's units are all demoralized (thus making them 7s), if side B attacks they would get a +1 for higher morale and +1 for all enemy units being demoralized (net +2).
 

On another question: The admonition that "Side includes undemoralized friendly leader and at least one non-AFV unit." for a +1 comes into question. Does that admonition mean consideration given to BOTH sides for the consideration or just the assaulting side?

This would be for both sides.

Consider this weirdness: 2 German INF in a hex, normal moral of 8, but are demoralized, thus morale of 7. They are attacked by Soviet militia infantry, morale of 6. The Germans would get a +1 for all units having superior moral. The Soviets would a +1 for all enemy units being demoralized.

1. You are correct that the higher moral involves only the units participating, which means that being being demoralized or reduced may affect one hex different than another. Don't forget that it also says "attacking" which would mean you only get the +1 when attacking, not defending, similar to how the attacking units in a fortification/town have a -2, but the defending units don't.
2. Initiative has no bearing on column shifts, although it may give you a chance to assault before the units defending can recover, giving you a +1 if they are demoralized.
3. They are separate modifiers, if you have a leader you can add other one for a 3 column shift. It's great to assault a hex with demoralized troops having a lower moral and your brave leader directing the action.
4. Yes both sides get the leadership bonus, but other shifts are dependent on whether attacking or defending such as "defending  higher elevation slope hex" or "All units of attacking side have higher morale"

The weirdness example would not give the Germans a bonus since they are defending, but the Russians would get the +1 for the German demoralization.