I bought a Development Programmer Board for ATtiny13A/ATtiny25/ATtiny85/ATtiny45SN and some ATTiny85 chips. I have tried combinations of boards with no luck. I would appreciate any help in what combination to use.
I also bought Digispark Kickstarter ATTINY85 General Micro USB Development Board BBC and was able to program it as Board: "Digispark (16mhz -No USB)" and Programmer: "USBtinyISP". I thought that the above programmer would be similar, but it doesn't work the same.
That board you bought is deceptively marketed. It is a board for use only with chips already loaded with the digispark bootloader. It is not an isp programmer and cannot be used to bootload a virgin chip.
If you bootload the chips appropriately first, it would work like the second board.
So, the thing to do is something like this to program the ATTiny85: https://www.instructables.com/id/Programming-the-ATTINY85-Chip/
Once the boot loader is loaded onto the main Arduino board, then the ATTiny85 can be programmed using "Sketch -> Upload", right?
Yes, that's the correct scheme for uploading code to an attiny85 the normal way.
Your terminology is wrong though - what needs to be loaded onto the main Arduino board is the Arduino as ISP sketch. It is not the bootloader (which is already present, and is how you can upload via the USB port on the Arduino board). A "bootloader" is a program on a microcontroller which runs only right after the chip is reset, and allows programming the microcontroller that the bootloader is running on. I would describe the Arduino as ISP sketch as a "programmer" sketch - it allows programming a part other than the one that the programmer sketch is running on.
Yes, I did use the wrong terminology.
I knew what I meant. That doesn't help others know what I meant though. 
Let me try again.
I load the ISP sketch into the Arduino, hook it up to the ATTiny85 as given then use the Arduino as a programmer for the ATTiny85. The sketch I load into the '85 can be the Digispark bootloader then I can use the subject board (see first post) to load additional sketches onto the '85.
Now, if I do that does the new sketch replace the '85 bootloader? If so, then I need to use the bootloader sketch as my base sketch then add my other code after that. That way I can always use the subject board to upload sketches. Correct?
Yes, essentially.
The digispark bootloader isn't a sketch, though (it's supplied as a compiled .hex file) - you select the digispark board in tools->board and do "burn bootloader" to load the bootloader.