The diagonal line sticking out of the target makes it asymmetric:
Per the Print and Cut doc:
Output is aligned relative to the center of the object or Group used as a registration mark.
Get rid of the diagonal line.
Even if you fix the targets, Print and Cut is the wrong hammer for what you’re doing.
A fixture to hold the blanks at precise locations, with the fixture precisely aligned to the platform, eliminates all the errors from trying to align things on the fly. Because the cats aren’t symmetric top-to-bottom, make the fixture with rectangular cutouts wide enough to hold the blanks in either orientation, aligned on their rectangular end opposite the cats (or soccer balls or whatever).
Then:
- Cut a stack of blanks from plywood, packing them tightly to reduce waste
- Set up / align the fixture
- Drop blanks into fixture
- Push them to align their rectangular end
- Burn front side
- Flip them over
- Push them to align their rectangular end
- Burn the back side
- Profit!
This takes a bit of planning to get everything laid out properly, but it eliminates all the error-prone fiddling at the laser.
The two sides probably have different characteristics, even if they’re “the same wood”. As a quick test, run identical Material Test patterns on both sides of one sheet.
That will ruin the focus and exact positioning, as well as blur the patterns. Make sure the machine is solid before you start the next time! 
If the speeds are in mm/s (typical for big CO₂ machines), then the machine was not traveling more than maybe 500 mm/s (again, typical) in either case.
You can set any speed you want in LightBurn, but the machine controller will limit the actual speed according to its configuration settings. Given that the controller configuration speed limit is much lower than your settings, the actual speed didn’t change at all.
For millimeter-size shapes like text, the actual speed will be much much much slower than you set, because the machine cannot accelerate to high speeds in short distances. You will get better results by “engraving” the designs as an image, because raster scanning runs at a much higher and well-controlled speed.
And it will be, but the challenge is to align the material and the pattern as you expect, then ensure the cut settings match the machine’s capabilities.
Fix the problems with the targets, speeds, hardware, and layout, then see if the situation improves.