thanks for stepping in - I don't characterize myself as Buddhist (in a religious sense) either, but after ~2 decades of Buddhist martial arts, the philosophy inevitably begins to creep in. what you say is the western interpretation of it: it kind of describes Buddhism, but doesn't quite do the idea right.
Buddhism, as I understand it, actually has one core principle, which leads to two main consequences.
the principle is: all separation (between parts of the "universe" - us, others, animals, objects...) is only an illusion. try to overcome that illusion and you overcome the idea of suffering; after all, your hand doesn't suffer from having to wash your butt daily, they're both part of the same body.
(1) let others do want they want, we're all the same. [this one is massively misunderstood: it doesn't mean be passive to other people's misdeeds or needs. quite the opposite. it means don't impose your views on others (there are too many viewpoints, and yours is not necessarily the only valid one), but you can and should act in consequence of what you believe is right. for examoke: don't preach to me about being honest, but do stop me from stealing or do expose me as a liar.]
(2) be compassionate with everything that feels (e.g. living things) - we're all the same. if one part hurts, the whole hurts, and that includes yourself.
and then of course there are various religious rituals and lots of philosophic decorations, but the above is the essence.