Basically the only difference in your two choices is wood vs. stone columns. I'm certain that stone will be noticeably more expensive. I believe you can buy plans for carports online at sites the sell house plans.

Best wishes!

Yes, I agree and that's why I have decided to go with the first option which I believe is called "Beech Wood Style Carport". Google search shows:

https://www.google.com/search?q=beech+wood+car+port

I still need to find, if I could get a pre-fabricated kit or how much do I need to budget for this project?

I did find carport blueprints. There are several in the style you are wanting.

https://www.thegarageplanshop.com/carport-plans/garage-plans/78/1.php

Hope this helps!

Those pictures are going to cost quite a bit of money given the wood type depending where you are. In the US, you could buy a kit from 84 Lumber to do this.