Brand an image with a deep set icon

[CtrlAltDel]

Hello everyone.  I always know that whatever problem I'm having with Gimp or whatever I need help with, it can be solved here.  I have lately started creating Linux Mint wallpapers and have so far created about 100 of them over a period of a year or so.  

One I have never been able to create though is one that looks like the LM icon is set deeply in, let's say, a colorful fence.  Sorry for not knowing the proper term for this or not having an example.  I would like to take the icon and make it look like it is about to burst through the fence, or whatever else I may use as a background.  Not from behind but, from the front.  You know, sunk deep into it, almost like a horse being branded and with it having some depth? is the best example I can think of at this moment. 

The image below is kind of blah.  I was hoping for something more than just changing the mode also, which is okay but not what I was looking to do.

[PixLab]

I would do first a Filters ➤ Map ➤ Displace because of the planks joins, then a Filters ➤ Map ➤ Bump map

Start by making your logo to Layer ➤ Layer to image size as well as your BG, it's very important that layers are at the same size, then
Duplicate your background, then the duplicated background, Colors ➤ Desaturate ➤ Desaturate default are good enough, then select your logo, once logo selected Filters ➤ Map ➤ Displace chose as Aux Input the desaturated background

   

Once the logo is done, duplicate it, then like the background ➤ make the duplicated logo de-saturated, then Filters ➤ Blur ➤ Gaussian blur (4 or 5 is enough)
now keep visible only the background original (colored) and select it in the layer stack, then go to Filters ➤ Map ➤ Bump map and select as Aux input your desaturated and slightly blured logo

   

If you want to keep a bit of color of your logo, you can re-activate / visible the logo displaced (which is still in color) and play with the layer blending mode and opacity (example below with Mode "Burn" I did not play with opacity below, but you should)

   

[Ofnuts]

Something like this perhaps? https://www.gimp-forum.net/Thread-Pressed-text

[CtrlAltDel]

Thanks a lot for the detailed explanation, PixLabs. That is very close to what I was wishing for and I'm about to give it a shot and see what I can come up with.

Thank you too, Ofnuts. That also is kind of what I wanted and I may try to combine yours and Pixlabs advice and create a amalgam using both of them. :-)