The masons lay claim to triangles as originating from pyramids, with the sun rising over the capstone at dawn. They equate Lucifer to the Sun and Horus, all 'angels' or 'gods' of light, where stars were once believed to be angels. Who invented the triangle? The first recognizable triangle would have been a mountain, so you can blame gravity. To me it means the Most Holy Trinity, and nothing else. It's only a shape, remember that.
As for two-headed eagles, yes they are fairly recent. Phoenixes as you know represent reincarnation which runs deep in masonry/occult.
But again, eagles, birds, lions and so forth are frequently symbols of Christianity too - and why not! However, if it comes to determining occult origins of symbolism, Freemasonry usually writes its language on the symbol by keeping to some kind of geometrical shape or pattern. For example, the symbolism of bees, often used in Anglican school crests, are chosen not merely to symbolize the efficient cooperation of workers for a common cause, or something like that. The bee represented an Egyptian goddess (Neith? i think), or the "Great Mother". Poor bees. The trick is that the proportions of the bee as drawn are made to follow certain patterns, like fitting the points of a pentagram, for example.
It is all quite interesting to study symbolism, but it is wrong to let symbols as such have a power over your thoughts. After all, visual images only exist to serve man, not to govern him. Let Christian symbolism be in your home, in your life, in your work. Recognizing those symbols in other people's work might give you an idea of who they are or how they think. Knowing about the symbolism of the occult is very useful because people who are entangled in all of that can't help but give themselves away in that regard. :detective: