It feels a bit like you tried to make the jump from basic addition to calculus without much in between - By that I mean that you were having difficulty getting Cut Shapes to work because you weren’t using the Shift key to add the cutting tool to your selection last, and a few other basic questions that I asked along the way were misunderstood or misinterpreted, which makes me think you’re still fairly new at this.
I mean no disrespect by this at all, but this is a reasonably complicated process even with the software assisting as much as it does. If you’re as new to this as it feels like you are, it would worth doing a couple of “throw-away” tests on cardboard just to get the process down before you dive in and do it on something big and important.
This is actually how I did it myself - I made a small one (only two parts, not three), using the same overall process as you see in the video, but on scrap material that I cared nothing about. I didn’t record the first try, because I was sure I’d mess it up. (I didn’t, but I was fully prepared to). The initial test did help me catch a couple small mistakes, and also gave me the confidence to do the bigger, 3 part version.
In the image you show above, the alignment marks didn’t engrave in the same places, and the 2nd part of the engraving is off by a significant amount. Without standing over your shoulder and watching what you did I can’t make any guesses as to where it went wrong. If you can post step-by-step pics or a video of what you did, we might be able to spot an error.