ATTinyCore is an excellent choice. I know some work has been done over the last year to add support for the DigiSpark.
As mentioned above, the DigiStump company that created the DigiSpark has gone AWOL (though my impression was that the cause was them spending 4 years on a failed attempt to pull off a 3D printer KickStarter). So the official boards platform for the DigiSpark recommended by the Jaycar instructions is outdated and abandoned. However, I see you found the fork of the boards platform ArminJo created. This is well maintained and up to date. So that is also a good option. If you find that for some reason ATTinyCore does not meet your needs (I have no reason to think it won't), then we can definitely help you to get the ArminJo platform working.
Yes. It is a really horrible situation with this terminology. The term "core" has traditionally been used in the Arduino world when referring to the C++ library that defines the Arduino Language API. For example, here is the one for the Arduino AVR Boards platform (e.g., Uno):
https://github.com/arduino/ArduinoCore-avr/tree/master/cores/arduino
But the term is also used for the entire collection of files that are used to add support for a group of boards to the Arduino development software. For example:
https://github.com/arduino/ArduinoCore-avr
Due to the ability to reference resources from other boards platforms, the latter type of "core" may or may not contain the former type of "core".
For this reason, an official decision was made that the term for the latter is now "platform". Unfortunately, a lot of people didn't get the memo and are still using the ambiguous "core" term when referring to Arduino boards platforms, including even some (but not all) Arduino employees! In order to more clearly differentiate the two, I refer to the former as "core library" and the latter as "boards platform", and avoid use of the term "core" by itself altogether except when disparaging the use of it by others.