I just skimmed the article, but it reminds me of something from my own PhD in fluid dynamics. I worked in liquid jets/sprays, basically, when a stream of liquid flows from a nozzle and breaks into droplets. Nearly everyone has seen this in some form.
I published a "regime diagram" improving the existing categorization of types of liquid jet breakup. There were a lot of changes from the status quo, but one that strikes me as particularly sad was the addition of a new regime that I called "turbulent Rayleigh". Every time men pee, they create a turbulent Rayleigh liquid jet. Yet this was a mostly foreign concept to spray researchers! You can find a few papers that identify what they view as an anomaly, but the papers seem to be mostly ignored and they don't go beyond saying something like "Something here is weird, future researchers should look into this". I did my share, making a theoretical model of the regime, showing how it's fundamentally different from the conventional laminar Rayleigh regime. Most spray researchers would consider a "Rayleigh" jet to be inherently laminar, but that's a misconception.
The reason why this happened is that liquid jet/sprays research is heavily biased towards fuel sprays, which rarely ever have this regime. You basically need a long tube (or something similar) to reduce the Reynolds number for turbulence to appear, which usually doesn't happen in fuel spray nozzles. Fuel sprays tend to have lower surface tension and higher viscosity than water/pee too, which makes seeing a turbulent Rayleigh jet even harder.
That's a problem of aim, not spray formation, and thus beyond my expertise. The Rayleigh regimes produce large droplets (mainly diameters similar to the orifice) and not a finer spray that could go everywhere.
Also: Peeing into only the bowl won't necessarily minimize the mess. Air gets "entrained" into the water and that creates splatter. Best practice is to target the porcelain inside the bowl (just above the water line) and adjust the distance so that the stream has broken up less as droplets will create more splatter than a solid stream.
The biggest one is that many guys are just vile. Sad but true. They don't even try to aim, because someone else will clean it; and they don't even wash their hands after.
The next biggest is that sometimes where you point isn't where the stream goes, whether from a random pinch in the tube or a proprioception error.
And sometimes the stream is split, and it's not entirely predictable when that will happen.
Spray?? Never had an issue like that. Can't speak for everyone though.
To add to the other responses: sometimes it's splash-back from the bowl water itself. It's gotta be toilet bowl dependent, and I assume some people are just too lazy to clean theirs up.
I published a "regime diagram" improving the existing categorization of types of liquid jet breakup. There were a lot of changes from the status quo, but one that strikes me as particularly sad was the addition of a new regime that I called "turbulent Rayleigh". Every time men pee, they create a turbulent Rayleigh liquid jet. Yet this was a mostly foreign concept to spray researchers! You can find a few papers that identify what they view as an anomaly, but the papers seem to be mostly ignored and they don't go beyond saying something like "Something here is weird, future researchers should look into this". I did my share, making a theoretical model of the regime, showing how it's fundamentally different from the conventional laminar Rayleigh regime. Most spray researchers would consider a "Rayleigh" jet to be inherently laminar, but that's a misconception.
The reason why this happened is that liquid jet/sprays research is heavily biased towards fuel sprays, which rarely ever have this regime. You basically need a long tube (or something similar) to reduce the Reynolds number for turbulence to appear, which usually doesn't happen in fuel spray nozzles. Fuel sprays tend to have lower surface tension and higher viscosity than water/pee too, which makes seeing a turbulent Rayleigh jet even harder.