They mostly run Windows or some variation of Linux, though it is feasible they might run Android, and you do run into the occasional Mac (usually some form of Hackintosh).

Back in the 80's, hardware was simple enough that one could feasibly run the computer without a separate OS underlying the game. Hardware these days has become complicated enough that writing your own OS is impractical, typically requiring hundreds of engineers writing device drivers and kernel code (and that's assuming you actually have access to the specs for devices like GPUs). So people use an out-of-the-box OS, and most of the games you seem in modern cabinet installations would run just fine on your home computer.

So, if it is really like developers building their own gaming computer (Buying and putting together their selected video card, motherboard, processor etc) then putting it inside of a cabinet connecting to arcade parts like the joysticks and running their own arcade game off of the hard drive...

Does that mean the games in those modern arcade cabinets are programs running on Windows? Or other Operating system like Linux, or maybe the developers own made distro of Linux?

Yes. There's a whole industry behind it. The Taito Type X2 is widely used today (BlazBlue fighting series comes to mind) and runs on Windows XP Embedded.

Basically what devs gain from this is a friendly known environment (i.e. x86 processor, Windows XP, DirectX, Visual Studio), stable known hardware specs (like in console development); and has a few customizations (most notably the hard drive is encrypted and all data in and out is decrypted/encrypted on the fly to prevent people from just cloning the drive and distributing the game online in pirate sites)

I worked in a company that built similar products (big boxes with buttons and a screen that you play "games" on). We had some that used Windows XP Embedded, and some that used Linux.

Absolutely not worth it to develop your own OS when other options already exist

You'd actually be surprised how many other devices around the place are using common OS's like Linux -- e.g. your network router is a very likely candidate.