The libusbK drivers are the ones you want. Traditionally the libusb-win32 drivers were used but I have found those don't work with the recent versions of the Arduino IDE (actually Arduino AVR Boards). If you want to try the other driver I'm sure it wouldn't hurt. I have used a program named Zadig to install either driver and it makes it very easy, and is actually recommended by the creator of the USBasp. I can provide more information on that if you like.
ArduToys:
The fact that it did work on the FTDI Programmer made me think that I should have been able to program with it??
The problem you're having is with the computer/AVRDUDE recognizing the USBasp, it's unrelated to anything happening on your Arduino board, so the ability to use the FTDI is irrelevant.
ArduToys:
My understanding is that the FTDI Programmer does not need a bootloader.
The FTDI DOES need a bootloader. When you upload using an FTDI you're sending the program via serial communication. You need some firmware running on the microcontroller on your Arduino board to handle that communication and then write the code to the flash memory.
ArduToys:
Does the USBASP always need a bootloader?
No, when you upload using an ISP programmer such as USBasp the process writing the code to flash is handled solely by the microcontroller hardware and the programmer so no bootloader is required to help with that process.
ArduToys:
Should I upload the bootloader every time before I try to upload with the USBASP?
When you do a Tools > Burn Bootloader the fuses on the microcontroller are set according to the Arduino IDE's hardware definitions for the board you currently have selected in the Tools > Board menu, then the bootloader is flashed to the board. So if you need to set fuses it can be useful to do a Burn Bootloader, even if you don't need the bootloader. When you do an Upload Using Programmer the bootloader is erased so, other than for setting fuses, there is no benefit to doing a Burn Bootloader before Upload Using Programmer. However, you should be aware that when you do an Upload Using Programmer the bootloader is erased so if you try to do a standard Upload using your FTDI after that it will fail. You would first need to do a Burn Bootloader using your USBasp before you could use the FTDI to upload again.
ArduToys:
I am using the "Programmer: USBASP" for my USBASP devices. Is that the right "Programmer"?
Yes