I've found what may be contradictory statements, so I'm looking for some clarity.

It looks like Careers are part of the path, so you can select what class you want to be if you start from the bottom down.  Or do you start at the motivation instead and just pick your career?  But then chart 1-1 says careers can be from any homeworld, but gives examples of norms.  If you use the chart with the careers, Rogue traders can't be from death worlds and astropaths can't be noble born, the chart just doesn't reach that far.

So are the careers part of the path or are they an open choice?

They are part of the path.  Using standard creation, a player can choose to start either with a Home World selection and advance downwards, or a Career selection and advance upwards.

GMs have the option to allow players a 'free' row where they don't have to follow the "directly above/below or adjacent to directly above/below" path rules.

You can also just let the player pick and choose everything as they like, if they have a strong character idea in mind.

The Origin Path is kind of a helpful how-to for newer players, and the GM can restrict their players more or less rigidly to it on a case-by-case basis, depending on how experienced they are roleplaying, how familiar they are with 40k, how solid an idea for a character they have, etc.