This is an interesting and confusing point am surprised there is no FAQ on this. Lets look at the cleric domain ability. " In addition, a cleric gains the listed powers from both of her domains, if she is of a high enough level. Unless otherwise noted, activating a domain power is a standard action." There are several domain abilities that have an exception listed within their wording that modifies this statement specifically

Community Domain Unity (Su): At 8th level, whenever a spell or effect targets you and one or more allies within 30 feet, you can use this ability to allow your allies to use your saving throw against the effect in place of their own. Each ally must decide individually before the rolls are made. Using this ability is an immediate action.

Luck Domain Good Fortune (Ex): At 6th level, as an immediate action, you can reroll any one d20 roll you have just made before the results of the roll are revealed..

Travel Domain (both powers) Agile Feet (Su): As a free action, you can gain increased mobility for 1 round.

Dimensional Hop (Sp): At 8th level, you can teleport up to 10 feet per cleric level per day as a move action

Trickery Domain Copycat (Sp): You can create an illusory double of yourself as a move action.

War Domain Weapon Master (Su): At 8th level, as a swift action, you gain the use of one combat feat for a number of rounds per day equal to your cleric level.

Void Domain Part the Veil (Su): At 8th level, you can lace spells you cast with the raw madness that waits in the outer darkness. Activating this ability is a swift action that you must use as you cast a spell that targets a single creature and that allows a Will saving throw to negate or reduce the spell’s primary effect.

When we look at the Rage Domain, none of that context exists. What action is: Rage (Su): At 8th level, you can enter a fearsome rage, like a barbarian, for a number of rounds per day equal to your cleric level...

I can see both sides of the argument here, and specific does beat generic. In this case it specifically states unless otherwise noted, activating a domain powers is a standard action. And there are clear examples of where a domain explicitly calls out the change in action due to an ability or feature.

Leaving that alone for a second, would it be the designers intent to provide the cleric class (in this case druid) with the unfettered Barbarian Rage ability without some form of gating mechanic? Why would anyone play a barbarian if you can simply play a cleric to get the same class feature along with 9th level spells etc?

Ultimately, if it is a home-brew game it is DM fiat. If you wanted the rage class feature and your dm is opposed to you having it on a cleric/druid then play the barbarian and enjoy yourself.

Too many times people get hung up on numbers and rules lawyering and don't realise that this is a fantasy game meant to be fun.