The term Orthodox is a social and cultural grouping; a way of describing a certain social group.
That is, we may share many characteristics, and those characteristics are useful to predict what members of the group do, or who will be a member of the group but none manage to define the group.
Being shomer shabbos publicly is not a sufficient characteristic. There are conservative jews who keep shabbos publicly. Keeping shabbos publicly is probably a necessary characteristic, but privately is not.
Being presumed a kosher eid is also not a necessary characteristic. I posit that I would consider someone orthodox even who was a known child molester–if he davened in my shul and sent his kids to my school etc.
It is just a social grouping; people are orthodox who identify and act as part of the group.
At some point, there could feasibly be a split among the group who currently is pretty much one large group, and it will become two separate social groupings. When that happens, probably one of them will be given a new name (I propose: Open Orthodox, to signify that they want to pretend to be Orthodox, but don’t feel bound by any of the Torah’s rules besides the social justice ones which they invented by pulling out of a horse’s tuches.)
So that’s why attempts to define are futile and irrelevant–the question is only whether they are considered part of the social group. I don’t know enough about the israeli politicians but Saul Lieberman (he should rot) was certainly not Orthodox even though he presumptively was better at keeping the torah than me besides for the dinnim of being a meisis umeidiach, which I am better at.