Did I damage the display coating on my MacBook Air ?

Hard to tell exactly from the screen shots. In one sense it does look like smudging, but that should be able to be wiped away with the clean dry cloth that came with your computer or other soft lint free cloth.

It also looks like there could be scratching if the cloth used was too abrasive.

Whoosh has received mixed reviews with some users claiming it caused smudging on the Mac.

For reference, here is the official Apple Support page on cleaning your screen.

Clean your Mac computer’s screen or display - Apple Support

If you still have the cloth that came with your MacBook, I would use that one instead. It may even be a good idea to purchase another one. Apple does sell the polishing cloth, but it is expensive. Since some whoosh users have said that it smudges the screen and your screen does look smudged, it may be worth the expense.

I'm sure your screen is fine and the coating is not damaged, since whoosh does not contain the harsh chemicals that would affect the coating. The smudging should go away with a dry lint free cloth, or the Polishing Cloth provided by Apple.

They have been included for as long as I can remember and were always in a little box that included Apple Stickers. I have purchased 4 Mac computers in the last 10 years and they all had the little microfiber cloth, but cannot say for certain all models included it in every year.

The Polishing Cloth sold by Apple is $19 (US) and a lot more durable than the small microfiber cloth that was included. The Polishing Cloth is expensive and I understand balking at the price, but there is nothing better and it was designed for cleaning the Mac screen. You can try less expensive ones, but I would avoid any that have a cleaner already on them.

I have only seen damage to the coating when harsh cleaning chemicals have been used or an abrasive cloth, which I don't believe is the situation you are experiencing. I have not seen where Whoosh has damaged the coating, only that it has caused smudging and I think the Polishing Cloth will solve that.

In my opinion, it would be worth the $19. It is durable and will last a long time. No cleaner is needed and can be used on all your devices.