No, but it will unlock from just (apparent) proximity which allows the thief enter the vehicle where they are then able to push the "Start" button. Again, the car will think the key is present and will start. At that point, the thief can stop the signal relay and the car will keep running without the key being "present".

It's not entirely clear from the article and conversation, but they're talking about passive FOBs that use the (presumed) proximity of the FOB to unlock the doors and allow the ignition button to work. These were created to stop you from having to physically dig out your keys.

If you need to push the button to unlock this won't work, but they could still smash a window and then relay attack the ignition. Challenge would be determining prior to the break-in if the FOB is close enough to relay. Plus the most vulnerable cars tend to be the most expensive. From the article I'm a little surprised the RAV4 has passive entry.

The thief just gets in the car and presses the button using the boosted key fob signal.

Yes, the Tesla Model S and X keyfobs do not require any button presses.

Not even without pressing a button on the car to start it?

Just the brake pedal.