Phono Cartridge for $1,000.

I think it means coil wire with a hollow core. I think,,,,

What is an "Air Core Coil" chakster? I’ve never heard that term.

The term is old
It's the opposite to an Iron Core

"Compared to the iron core type, which is the mainstream of MC cartridges, the air core type is not affected by magnetic distortion, so a natural tone color and precise sound field expression are possible" .

Another Air Core Coil type is an old Fidelity-Research FR-7fz for example 

I am a huge fan of Audio Technica cartridges.  I have not owned any of their MC designs as I have been a long time user of their very excellent top of the line MM, the AT20ss and it's still one of the best sounding carts out there.  I recently picked up an Ortofon A90 MC which of course is way more money than any of the items being discussed here, but this is a truly outstanding piece.   If I had to send my A90 out for service, I'd be very pleased with swapping back in my AT20ss!   The bottom line here is that if you go for an AT MC design, I am sure that you would be very pleased with the outcome.

It is the former the coil is wound around. That can be plastic, paper, aluminum, iron ...

... or no former at all - "air core"!