Loosing details and weird behavior on custom armor piece

[proc24]

Hey there, I am trying to go a little bit further and add a bit of shoulder armor to my orc, but Im afraid im having a bit of a rough time making it work so I was hoping someone does have a suggestion for me.

Heres the piece of armor I would like to achieve:

Its the shoulder armor from the course logo.

I started off with a plane and added some loop cuts into insetting the complete plane a bit for the outer armor part thats a bit thicker, I mean the outline:

Ok up until this part everything goes as planned, but when I add the shrinkwrap it behaves a bit weirdly:

Now it doesnt wrap as i intended it to be. Whats more is that when I add the subdivision modifier i loose a lot of detail or sharpness on the armor outline:

Then also the solidifier kind of doesnt work too

Finally after adding shade smooth it looks like he is wearing a shoulder carpet rather than an armor piece.

Edit: While posting the question it occured to me that this weird shrinkwrap is because of the rough shoulder skin I added with the alpha brush. I then tried to remodel it to keep the comparision between the new shoulder armor and the old shoulder armor, omitting shrinkwrap and placing it manually. It went better than before. I am still however loosing sharpness on the outline of the armor after adding the subdivision modifier. I tried fixing it with loopcuts to add an edge closer to the original edge to make it more sharp, but it didnt really work. I deactivated merge vertices so the loopcuts dont merge as i have placed them really close to the outline side but still no luck:

Can you give me some tipps on what to do different. Alternatively, can you outline the process to get to the armor im trying to achieve so I can redo the whole thing?

[NP5]

Subdivision does that.
You can add control loops to sharpen up corners. Or select the corner edges and use the Mean Crease in the N panel Item tab.

Below Green edge loops are control loops nestled up to the corner edges. (Colour is used here to highlight them, it is not standard Blender colouring and has a completely different use)
Red edge loop is the corner loop with Mean Crease applied. Normal for Blender to mark them red.

[proc24]

Thank you so much for your help!

I did add the supporting loop cuts, but it somehow didnt achieve what I was aiming for. Then it hit me. I may have been setting up my modifiers incorrectly here. So I checked and here we go! Subdivision surface was set to 1 in the viewport. Well of course then it would render the edges weirdly. Now after changing it to 2 as in the render settings the armor looks just as I intended it to be . I dropped the shrinkwrap modifier as stated above.

I did however not know about the Mean Crease option yet, so I had to give it a try and this too looks nicely:

Whats more is you end up with less edges here. Is there a downside to either approach? Especially when exporting them to game engines sometime. Im afraid I havent reached the course part yet where we do that.

Again, thank you for taking the time. Learned something new!