Hi Ulmicola and welcome to the guild. The bottom map is a lot like the top map but with shading applied. So assuming you have the colour all in the bag your really looking at how to shade that map so that it appears to have darker regions on the bottom parts of map hills and lighter parts on the upper sides of the map hills.
You can apply the shading using a greyscale shading map on top of the colour map using the arithmetic function and the best is use multiplication. So you have a colour map on the base layer and then the greyscale on a layer above layered using the multiply layer mode.
Where the greyscale is black the map will be black, where it is white it will be exactly the colour of the underlying base colour map. Where it is grey then it will darken the colour map.
So how to obtain this greyscale map is the next problem. That is most easily done if you create a height map of the region. The height map is a pixel based map where the height values of the region are mapped to pixel cells. To get that shaded into a greyscale its easiest with some software and many of them will do it. There is Wilbur which the member Waldronate can give advice on or maybe QGis will be able to do that too. Depending on the format of the heightmap you might be able to use a 3D mesh app like Meshtool or even Blender, but the specific mapping apps make the process easier than a 3D modelling app. It depends on what you are familiar with really.
Its easier to paint the heights and then generate the shading greyscale rather than to try and hand paint the greyscale directly.
If your not too fussed about the detail of the shading - such as in the map above - you can draw in the contour lines and then generate the height map from contours. I have a few posts explaining how I do that. Do a search here using my handle and Zebra as a keyword and they should show up. That can get you the height map but you still need to shade the height map to the greyscale.
But once you have the greyscale its easy. Getting the shaded greyscale from the height map is not too bad. Getting the height map from the map is not too bad if its not too detailed but you need to know how to do it.
Incidentally, the second image that you have posted shows heights of the subsea as well as the heights above sea level. Normally you dont shade them because the sun does not light up the subsea terrain. Normally the shading is more related to the depth of the water because the sea absorbs light and so more depth means gets darker. So for a better map than the one shown you should shade the above sea level and the subsea bathymetry using a different method.
I hope I haven't misunderstood the problem that you are having.
(Edit - link to my contour method:
https://www.cartographersguild.com/s...ad.php?t=52028,
https://www.cartographersguild.com/s...ad.php?t=33666 )