Your original code is good. It worked on the Uno. You just have the wiring wrong because the pins are different on the attiny.
Use this diagram to help you check. Its the blue Arduino pin numbers that you need to get right in your sketch.

Compare them to the pin numbers on these lines:
// mapping segment DP,G,F,E,D,C,B,A data pin
const int segmentPins[8] = { 7,6,5,4,3,2,1,0 };
// mapping digit 1,2,3 data pin, skip pin 13 on board LED
const int digitPins[nbrDigits] = { 8,9,10 };
If they don't match up, either change the numbers in the arrays, or change the physical wiring to match.