You would have to split the strip using Shift+K, then ...
just grab the right end of the left strip with your cursor and move it to the right. Then the last pic will be shown as long as you dragged the right edge.
You are trying to edit a curve, not a mesh object.
They are different kind of objects, the menus change depending on the context. Curves have no vertices or faces.
Take a minute to read the manual (not a video) to understand the difference.
I would go about it something like this - it's not "parametric" per se, but it's rather easy to implement so it might be a happy trade-off. Since you only need a small influence to the "dent", I would recommend omitting the subtracting of faces and shrink-wrapping, and instead use Proportional Editing.
First step is to isolate the section ...
Select only two points where you want to subdivide, and select Stroke > Subdivide.
Or just right click on the context menu and select subdivide.
Note that you can modify the settings for the tools and the post processing for the draw tools to get better detail and optimize how control points are distributed on the drawn lines.
For more information read ...
Make the bottom shape like this
Select this edge loop
Hit ctrl-b to create a bevel and play with the Shape slider
You can turn on Clamp Overlap and adjust the Width to get it to go all the way to the outer edges
Guess what... im holding another Major L, @FFeller just told me the solution, i thought it wasnt gonna work cause i just did so much other stuff to figure this out so i didnt thought this small thing can be the fix, but it was. Just pressed Alt + H. Veeerry weird why it didnt work earlier before i posted this question, cause i obviously have tried pressing ...
There's probably a far more elegant way than this (I'm not that experienced with the mesh editing tools) but I would simply select that middle edge and E to extrude that down to reform that middle face. Then simply select each set of 4 vertices, hitting F to create each of the side faces.