How to cut floor joist for stairway headroom?
That's a pretty tall order, you are into structural and would, in most areas, need a engineer or architect to sign off. It would almost certainly involve pouring a concrete footing for the post to rest on along with doubling some of the joists and using joist hangars with the proper structural nails. It's not super difficult but the amount of work required to do the job properly gets exponentially bigger as you add elements to the job. Cutting the stairwell into an L shape adds a lot of work. It's definitely the right thing to do looking at the picture. But it's not as easy as adding a post. 

Here's an example of an L shaped stairwell


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This doesn't necessarily apply to you, it depends on where the load points are above. An experienced contractor could easily figure it out but they is little you can do if something goes wrong. With a architectural stamp you have someone liable for any failures. 

Just to throw out an idea



Red is a post

Blue is a double 

Green is a cut joist 

The post could potentially be placed further right but it depends on how long the stairwell is and the code in your area.