It's one of those instances where definitions, especially concise ones, always end up conveying less meaning the the original word. They are really a bunch of behaviors or characteristics we associate with the words. What is classy, sexy, tragic, trashy etc.
What we can find are behaviors and characteristics that can act as a test for bureaucratic or political because its hard to imagine a company embodying them without embodying some others.
Political is neither equitable nor meritocratic. If a decision is made for neither of those reasons, it will probably be described as political. Applying it to hiring, promotions, raises, partnerships and sales seems to hold up well. If the beauty queen isn't the prettiest and there aren't any equity issues involved, it's political. If a hire isn't the best or a raise isn't fair, it's probably political. I think there's a definition here. It's interesting that this defines political decisions as pretty much a particular class of wrong decisions.
Bureaucratic is when rules are obeyed for their own sake, not for whatever reason they were made. I think it also implies lots of rules.
With those two definitions it seems like political is always bad. Bureaucratic might have some uses.