I would get hold of a copy of Keith Rowley’s book Woodturning A Foundation Course. Very informative, talks about all the tools and gives you some great projects to practice on. I still use the grinding jig I made following his design. A quick search returned copies available at £3.60 and £5 so not a big outlay…
As for the Lumberjack lathe you mentioned, I don’t know it personally, but it appears to be very similar to a Perform CCSL lathe, which I’ve had for a long time and still use (originally purchased from Axminster Tools). It was a great introduction to turning and I’ve made all sorts on it, but it is limited if you start to get into bowl turning and/or want to do bigger things. I ended up getting the bigger version which offers much greater flexibility.
If you can stomach the extra cost, the AW305 has some useful features over the Lumberjack lathe. I would say don’t underestimate the usefulness of the handles at each end. You’ll invariable end up moving a small lathe around and proper handles can make all the difference between a safe move and a clumsy/painful/costly mistake…
The really big feature though would be the variable speed control. The lumberjack lathe you’ll have to move the belt, but if it’s even a slight faff, there may be a tendency just to ‘get on with it’ and not worry about adjusting to the correct speed for what you’re doing. Besides the risk element, it may also make it harder to learn if you’re trying to turn something at the wrong speed (you’ll be more likely to get a catch). I’m not for a moment suggesting you’ll do anything unwise, but just reflecting on my own journey.
I’ve also found Axminster’s customer service to be very friendly and incredibly helpful any time I’ve had a query on something.
Whatever you decide, I wish you luck and hope you enjoy it. I started out woodturning to repair some old chisel handles and it opened up a wonderful and immensely enjoyable pastime!