Dishwasher air gap problem

There is a very simple answer to your question. Your dishwasher is draining into the side drain adapter on your garbage disposer. this was a common practice for many years. All the water draining from your dishwasher is being pumped into the upper basin side of the disposer. Solids from the dishwasher are blocking the drain in the disposer, so when you turn on the disposer, it grinds the waste and drains the water. This is actually the way it is suppose to work. Current codes require a separate trap for the dishwasher and not plumbed into the disposer. If you look at the side of your disposer, you will see the hose from the washer attached there. Only fix is to install a new separate trap and drain for the washer. Good luck.

I have had my Bosch dishwasher for about a year using the same setup you contemplate, so I'm familiar with the exact drain connector you have on the dishwasher hose. On mine, there was something of a taper between the 1/2 and 3/4 section and being rather flexible rubber, and I had no difficulty stretching the 1/2 section of the fitting to fit on the 5/8 gap fitting. I then tightened down the hose clamp with a hex bit on my cordless driver. The clamp was tight enough to somewhat compress a groove into the rubber, and I tugged on the thing as hard as I could. It isn't coming off IMHO. I'm at work, but can send a picture later if you need.

I understand you are worried somewhat about the quality of the connection. You could get a 1/2 to 5/8 adapter like this:

But that would entail 3 hose clamps and a section of 5/8 ID hose to connect between the adapter and air gap. That seems like more points of failure. At any rate, I think you already have what you need to get this working.