HATMAN
If dyitto was 50% on the way to the correct answer, you were about 66.67%.
Under your constraint you will not get a unique solution. As a matter of fact it would give you about 16 solutions from my initial givens. And no the solution you found was not the correct solution.
So you need to
look harder for the correct variant rule!
Just to save you all from wasting more time, the property spotted by both dyitto and HATMAN was not part of the variant rule. Even if it features in the correct solution, it would be there as a result/by-product rather than a cause/factor.
Hint 1: the variant rule can be expressed in very few letters, and I will show the initials in the following spoiler box:
Hint 2: did it occur to anyone that if one actually prints out the puzzle image and attempt to solve it on the actual paper, it might be easier to spot the underlying logic? (I even put it in Christmas colours for the holiday flavour!)