I suspect that Gramps' garden is better lit at night than the middle of a wood in the countryside. In which case, the IR illuminators on your trail cam will be pretty much the only source of light, so the spread pattern of the lights will be more apparent (brighter in the middle/central hot spot). Your first video shows this quite clearly, and the fact the central beam is pointing at vegetation that's fairly close, emphasises this effect. It also looks like you're getting some reflection off leaves (shiny beech leaves perhaps?) in front of and slightly above the camera - out of frame but reflecting the light back at the camera lens. This would explain the appearance of veiling lens flare in the upper part of the frame.
When setting the camera try to make sure there aren't any overhanging leaves, etc. in front of and slightly above the camera. Perhaps try lowering the position of the camera so it's not pointing down at the ground, but aiming parallel to it, so the concentrated central beam isn't highlighting vegetation within fairly close range of the camera, particularly any that is in the centre of the frame, where the beam pattern is brightest. It's a case of trying to illuminate the scene as evenly as you can. If there's a central bright 'hot spot' then the camera will close its aperture to try correct the exposure, resulting in darker edges to the image and blown central hotspot.
This is easier said than done in undulating terrain, so it's a case of experimenting to find the best camera position and remembering what you did if you get good results. Don't set the camera too low though (below about 18 inches), or you'll probably find it mists up with dew more regularly (depending on weather conditions). Also, make sure the lens is clean as there's a lot of dust and tree pollen about at the moment with the dry weather - blower brush time!
Your second video was captured when light conditions in the wood were fairly even, you can just about see the central beam of the IR illuminators, but it looks like there was probably about enough natural light for the camera not to actually need it's illuminators. Hence the even lighting across the frame and out well beyond the range of the IR illuminators. Hope this is useful and best of luck.