The rephrasing doesn't change things.
The fact that there are no core qualities is a real problem.
The fact that I cannot provide you an answer is a symptom of that.
In fact, I put that very challenge before a Straight Dope thread several years ago, and even the believers could not come up with a single agreed-upon quality.
So no, this is not a mere article of faith with me.
Okay, so lets take the four major religions - abrahamic, Buddhist, taoist, and hindu, and lets see what we can get. Remember, believers are not the best people to ask for unbiased judges on their religion.
First, creation. Buddha is a part of creation, as are those who attained enlightenment, the Jade Emporer was the head of the committee that created the world, JHVH did create the world, as did Brahma
What do they have in common? They all have the capacity to, or by their existence, create the universe.
Now, on the issue of power. Buddha has power over reality, but this world is transitory - its our job to enlighten ourselves out of it. Jade Emporer can do a lot, but has committees to deal with. JHVH is constrained by the fact that his every action must be good.
What do they have in common? Well, each one has virtually unlimited power (so close as to make no difference) inside our universe, even though they may be restrained by extra-universal constraints (other lesser Gods, their omni-benevolence, whatever).
How about knowledge? Well certainly the knowledge for the polytheistic religions isn't perfect. But they can get any piece of knowledge they need through lesser deities. So we gain our third criteria - can obtain any knowledge within the universe.
How about their motivations? JHVH wants to redeem souls, the Hindu Gods all have their own motivations, Buddha is enlightening people to the true (i.e. not our) universe, and Jade Emporer has a zillion little things. Fourth Criteria - has goals outside our universe.
Lifespan? They all agree - the concept is either meaningless, or longer than the universe.
So what do we have?
An immortal being or beings, who have ultimate power within our universe, even if they are constrained by extra-universal forces or internal limitations. They have access to any knowledge they need from within the universe. They have goals and ambitions that are related to the universe, but exist outside of the normal scope.
Is there a reason this definition is unacceptable?