Why does this circuit give different values for different methods
In your 1st attempt you have one of the legs of the 20R resistor connected to b, which is incorrect. Your 2nd diagram is correct.
In your 2nd attempt on the last line of your calculations you have ... +1/2 = ... but that should be 1/6 as you calculated 6R on the previous line.
After that you still need to add the last 8R resistor, and you're done.