Every establishment has different rule so you should call every place you plan on visiting. That’s probably why you’re having a hard time finding concrete info. At my uncle’s restaurant you just need a picture of a test. <deleted> Some smaller establishments will only accept proof of vaccination and won’t accept any negative test at all.
As far as I know, the city/state has stopped distributing free tests. I was able to get some and we received the Quickvue test so I’d expect most places will accept that brand. Binax is also popular in Hawaii.
Also, it was difficult to find at home tests in stock for most of September and early October, so I would suggest bringing some from home. The stores have plenty of tests at the moment, but supply chain issues + tiny island in the middle of the ocean ... you just never know.