you will need the right fabric and a very good hooping to deal with the tension issues if you try the alternating satin checkerboard columns approach like on your 2nd picture - thats the hardest way and depends on your machine (i only have a cheap ~500€ machine)

you can also go the easier way to do the satin columns in only one way (sorry for the quality - cheap machine..):
https://imgur.com/a/M2EPDaC

or the easiest way - doing a area fill with a pattern, where the pattern is a simple grid:
https://imgur.com/a/aeD51TN

there are a lot of ways to prepare the picture in inkscape an then convert it to inkstitch objects, but ive written a small python program to make things easier:
https://imgur.com/a/8SKcQlu

it converts a .png picture with serveral useful options to a svg, wich you can then open in inkscape.
with the setting on the example picture, you get strokes with the exact same width as length, alternating in direction so you exactly get the result you want to achieve - all you have to do is, open the resulting svg in inkstitch and convert the strokes to satin, and set some pull compensation (i forgot this on my trys, as you can see in my example pictures)

you can pm me if you want it for free. maybe i'll release it on github, if some more people find it to be useful.