As you have found out, filters can cause issues.
Every time light hits a reflective/transparent surface some light is refracted through the surface and some light is reflected off of the surface. The coatings on lenses and filters are meant to suppress those reflections.
Single-coated is better than uncoated, and multi-coated is better than single. The light source (direction/size/etc) does affect this, and you can think of multi-coated as having a multi-directional coating... i.e. more effective in more situations.
The issue could be the lens coating, the filter coating, or a combination of the two. But a better filter (or no filter) will certainly be better than what you've got going on.
The first diagram shows this effect with a filter showing the reflected component at both surfaces, and then with a filter/lens interaction (I didn't add the secondary surface reflected components for the lens element). The secondary reflected light path travels a slightly greater distance; therefore it has a different focal distance and it will focus slightly in front of the primary light path.
This is exactly the same effect as focusing on a reflection in a mirror... the focus distance is not the distance to the mirror, and it is not the distance from you to the object; instead it is the distance from you to the mirror plus the distance from the mirror to the object being reflected. I.e. The focus distance is the total distance the light has to travel.
The result of this secondary light path being focused slightly in front of the primary light path (short of the sensor/image plane) is a loss of image sharpness/detail/contrast and nisen bokeh... the double line bokeh shown in the horse image. You are basically recording two images stacked, and one is slightly out of focus.
The takeaway should be to avoid using filters when they are not needed. Add-on optics (filters, diopters, TC's, etc) are not part of the original lens design; and therefore there can be no correction incorporated into the lens design for any optical errors they add... they (almost?) always adversely affect IQ to some degree.
But sometimes the tradeoffs of using add-on optics/filters are worthwhile; or even necessary. So if you are going to use a filter, avoid using really cheap ones (uncoated/single coated).
However, you do not always get what you pay for with filters; except in the case of very cheap ones, those are almost always bad/worse. And a filter can also introduce its' own optical errors; and you can get a bad copy of one as well. Roger Cicala did a nice review/test of 20 protective filters (UV and clear) which shows the variability/issues: https://www.lensrentals.com/blog/2017/06/the-comprehensive-ranking-of-the-major-uv-filters-on-the-market/