Plumbing – I would like to replace air gap in the sink with a flat plate, what is that plate called

I'd like to thank everyone participating and up-voting. I went to home depot today and bought a new air gap. Upon a further inspection I found out that my air gap is missing a cover: a piece of plastic that routes the water coming from the DW into the drain.

Long story short, it somehow got popped out and my wife "saved" it by sticking into the kitchen drawer.

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.