The Starbucks manager didn't ask the men to leave, or tell them they had to order something if they wanted to stay, or anything like that. She called the cops without giving any indication there was a problem. That's why the men and the other customers were so astonished. They had no idea why the cops were there or why the men were being put in handcuffs.
Even if the Waffle House and Starbucks managers, acting on behalf of the owners, had asked their customers to leave, that alone isn't sufficient for the police to make an arrest. The police can't just violate your civil liberties on an informal, possibly offhand comment. The business owner or the police have to give the customers an actual Trespass Notice (verbally or in writing but one that can be formally filed) and the customers must appear to be violating it before there's probable cause to make an arrest for trespassing.
Except that the cops did come and tell them that they needed to leave, they argued for several minutes and then were arrested when they refused to leave. That's the verbal warning to leave the property.
Cops can't just walk up to you and say "get out" and then arrest you if you don't immediately comply. They can threaten you with a citation that will require you to appear in court and might result in a fine, but they can't just haul you out in handcuffs. You have civil rights and civil liberties that the cops are
not allowed to violate.
For the cops to act lawfully in removing a customer from the premises, a Notice of Trespass has to be given. Not in an informal, casual "you have to leave now" kind of way, but as an official notice. The second requirement is probable cause. There must be probable cause to believe a person who has been given a notice of trespass is currently violating it before an arrest for violating a notice of trespass can be made.
You guys arguing that citizens must immediately comply with police officers or be arrested on the spot are pretty scary. Not because I fear you personally, but because your ideas about the powers of police officers to incarcerate citizens are like something out of a Franz Kafka novel.
ETA: I once saw a police officer giving a group of people a Notice of Trespass. He very clearly explained to them what it was and what it meant. The cop wrote out a citation, handed it to the individual who had invited the others into what might not have been his property, and told him how he could get the notice dismissed by appearing in court with proof of ownership. The group dispersed and the guy who got the notice left the premises. It was all very polite and in accordance with the law, exactly how the incidents at Starbucks, the Waffle House, and the United flight should have been.