There are many code requirements for stairs. Make sure one decision does not cause problems in other requirements. The stability of the newel post is very important.

Attaching to the stringer and floor will be a sturdier support than the floor alone.

Note metal runs, discarded on stairs and wall. (a worthy dismantling)

However, if the pickets are attached to a continuous metal run under the railing, your choice would be well supported.

Here's another way to provide support.

my stringer is not open so

This will be how it will look in my case..

what would involve to make it open staircase like this photo below?

I would go with the more stable option as long as it meets code - the visual appeal isn't much different between the two options.

#1 is the way to go. Sturdier and ties in with the stairs (visually) better.

If possible the post should go through the floor and tie into the structure below.

If your finished flooring is installed and you choose PPF.' solution, remember: "measure twice, cut once."

what would involve to make it open staircase like this photo below?

Complete new stringers and steps.

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Most likely, the contractor will use a newel bolt anchor like this.

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For a really solid railing, be sure that the balusters are connected to the handrail with the metal run I posted pics of before. (Can't remember the exact name!)

The railings in our current pre-owned home don't include them and I fear if I fell against them they'd collapse. And I'm well-below the average American male weight of 197.9 lbs.