It matters when one of your goals is to determine what the people around you believe.

Either an assertion is a conventional belief, which is a truth that has no evidence, or a logical conclusion belief, which they don't fully understand (eg lack of understanding about cause and effect in a study). These concepts are often confused, but neither of them are fundamental to a rational discussion.

A lack of acknowledgement when these biases are discovered, is a compelling reason to eject from a discussion. Discussion is one tool to convert beliefs to truths. Sometimes the beliefs changed are ill-recieved, but it your own or another's.

>> Most of the time it doesn't even matter whether that someone actually believes what they say.

> It matters when one of your goals is to determine what the people around you believe.

What people say is some of the lowest fidelity information to use when trying to assess what people believe.

Actions >>> Words

Body language is very important and sits between Action and Words.

It should not. The decision to assess a claim in context (eg a speaker's expertise or lack thereof) or in isolation is often an important one, and creating an implicit contextual fork and then attacking an opponent for choosing the 'wrong' prong is a well-worn rhetorical tactic for seizing control of an argument on procedural rather than substantive grounds.

It is important what people believe because this is what will influence their actions.