Am I wrong?

Bougainvillea is a climbing plant, not hanging plant. But it does not flower on the primary climbing limbs, it, like most vines, flowers on new green shoots. These can go up or down depending on the situation.

You are correct in my view that this depiction is faulty as it does appear to show flowers emerging from a twisting, climbing stalk. In fact the branches with flowers exist for the primary purpose of flowering, not climbing.

I did not think that tattoo was Bougainvillea for that reason. The flowers in the image are also fairly evenly distributed where in reality they tend to cluster.

I think you are broadly correct that the image is faulty. Bougainvillea is in fact in nature a climbing plant. It can be grown as a shrub in which case the flowering branches are more likely to hang.

On a climbing plant the flowering branches are like to go up vertically when they emerge and hang later as they gain weight.

It is a beautiful tree and flower and would make a great tattoo. There should be a lot of images on Google that will help.