From your other question I know that the lat pulldown machine broke down, however:
Using a machine does not really prepare you for the real exercise. To improve any body weight or free weight exercise it is best to not use a machine. The machine controls the path of the weight and you don't have to worry about the weight dropping uncontrolled or tipping over to the side. This means that the muscles that would normally be used for this, wont get a lot of training. However these are crucial to perform the real exercise with the same weight.
Also if you are trying to do one pull up, why are you doing eight repetitions on the machine? I'd go for a less reps, more sets and higher weights. Maybe something in the 5x3 range, however this is purely based on my personal approach to arranging workouts.
Why is it harder to do pull ups on the straps, compared to the bar?
It's somewhat comparable to how the machine is to a real pull up. With a bar your hands have a fixed point in space to attach to, with those straps they don't. Your whole body has to tense far more to not go in a swinging motion, to not twirl etc.
If you really are bound to the straps (haha, pun not really intended) I'd try to attach them as closely as possible to the ceiling or the wall to reduce their movement. I don't own those straps, I've only have just seen them on youtube, so I am not sure how to actually do that.
Maybe you should check for a pull up bar again. Mine was relatively cheap, about 30€ and ceiling mounted. You don't need a lot of space for it.
Thats the Home-Gym part of my room, directly after I drilled the bar against the ceiling. My apartment is really small (~20m²), it still works out fine. I can no longer fully open the window, but whatever…
As a side note: I also did pull ups hanging down the locker on the left before I had the bar. Totally uncomfortable grip, a bit dangerous, especially when the locker tips over, but you are not able to cheat by throwing your legs up.