You are wrong. "Within 6 inches" includes "6 inches away". Here is the relevant part from the Core Rules:
"[...]So, for example, a unit is within 12" of another unit as long as any model from one unit is 12" or less from any model from the other unit[...]"
(https://www.games-workshop.com/resources/PDF/AoS_Rulesheets/ENG_AoSSW_Rules_booklet_web.pdf, page 2)
You are absolutely right about this. Even without the actual rules at hand, you provided an intuitive and elegant explanation to the problem. Another person voting you down without even discussing your argument shows that you left no real room for a counter argument.
Your intuition is correct. If an "Ambush" ability states that the unit has to be wholly within 6 inches of the battlefield edge, the following Designers' Commentary clarifies the meaning of it:
"[...]A: A model is wholly within a certain distance if every part of its base is within the stated distance[...]"
(https://www.warhammer-community.com/wp-content/uploads/2019/07/8f9bd00c.pdf, page 3)
In combination with the quotation about "within" above we can see that "wholly within 6 inches of the battlefield edge" means that one point of the models circular base can be exactly 6 inches away from the battlefield edge. Lets put the model on a line with the objective and let the line be perpendicular to the battlefield edge. Since in your example the center of the objective is 12 inches away from the battlefield edge, it means that it is now exatly 6 inches away from the "6 inch point" of the models base we discussed beforehand. Since "exaclty 6 inches away" is included in "within 6 inches", the model is now within "capturing distance" of the objective.
I am sorry that multiple people were trying to tell you that you are wrong and did so without any backup from the rules. We should always be wary when people speak from intuition alone and fail or not even try to connect their statement with the actual rules of the game.
I hope my explanation is clear and that you can make use of this knowledge during your future games.