If you flag down a vehicle, especially a taxi, you wave at it as a signal for the driver to stop. They flagged down a passing family who stopped to help them. [
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Look for the light – if the roof light on top of the cab is lit, the taxi is available for service. Raise your arm to “flag” the cab, and do your best to make eye contact with the driver. Be prepared to let the driver know where you're headed.
The best thing to do is give them a visual point of reference where to stop. It is not rude to want to be dropped off at an exact spot. Remember the driver doesn't know you and your routine so they rely on you to say where to go. Polite would be giving them time to safely move and stop where you would like to be.
But for the most part, a cab company owns the vehicle. They lease it to their drivers who in turn get to keep 100% of the fares and tips (some companies charge less for the lease, but retain a portion of the fares).