Would you confirm that the expressions below share only ONE meaning in which they may be interchangeable (YES/NO):
on a run
on the run
on a run - 1. running: Colonel Armant came on a run to find out who gave the order to fire.
2. (of a dog) on a leash: My parents live in rural yet townish sort of area - we had a dog we kept on a run.
3. (of a dog) exercise in running: Likes to go for long walks in the park and good for anyone who wants to take a dog on a run.
4. on a single trip or journey: They were the quickest two-wheel drive car G2 and G5 combined, and posted the quickest time on a run in the same category.
on the run - 1. = on a run 1: Workers roofing a house nearby heard the screams and came on the run.
2 without stopping or pausing: We are so busy at the office these days that I have to eat my lunch on the run.
3. busy moving from one task to another: With three young children to take care of, Helen is kept on the run every minute of the day.
4. running away from prison, the police, etc.: He went to meet a Franc-Tireur who was on the run after killing several Germans.
5. (of an army) retreating: On the southern section of the front we now have the enemy on the run.
6. severely defeated: Conservatives are on the run throughout Wales.