If you look at the original design with separate resistors, when Q0 switches high, the anode side of R1, the first 1K resistor on the first LED, will go from around zero to around 2 volts depending on the LED, and none of the other LEDs are affected which is what you want .
In the single resistor version when Q0 goes high the anode of the 1st LED goes close to Vcc and the cathode, and all the other cathodes, goes to around Vcc - about 2 volts, depending on the LED, and this can upset the cathode voltage the other LEDs see when they try to turn on. Use the first version with separate resistors.
This post was edited because mikej spotted an obvious mistake.