I see that some place the beams/joists on top of the front load-bearing beam and some attach them to the load-bearing beam with joist hangers.
Is there any difference in these implementations? In my view, it feels safer for the load when the joists rest on the load-bearing beam.
Which method is correct?
Is there any difference in these implementations? In my view, it feels safer for the load when the joists rest on the load-bearing beam.
Which method is correct?
Both methods are correct. It's more a matter of preference. The advantage of attaching the studs to the joist with joist hangers is that you get a more streamlined construction. If you've chosen the right joist hanger and screwed/nailed for the correct load, that method works well 
Yes, using a ledger is primarily cheaper, simpler, and faster, which is why it's often used if there's enough space. However, it increases the height significantly more, so if you want or need a lower solution, joist hangers are a good option. As long as it's done correctly, both solutions are equally durable.