It's one thing to stand your ground when you're at home or even in a public place. But when you are in a place that is owned and operated by someone else (your place of employment, a restaurant or movie theater, a passenger aircraft, etc.) you have to adopt a completely different mindset. If a person representing the owner asks you to leave, it's best to get up and leave even if you think you're right in that particular situation.
Read the fine print of your ticket to see what rights you voluntarily agree to give up to the airlines, in addition to the right to be treated fairly by them.
Given my experience, the airlines employees are allowed to lie to customers, even when it’s obvious they are doing so.