I see some units have a stationary splitter and some have a stationary block or stop. Is there an advantage either way?
As others have noted, the horizontal/vertical splitters are always stationary block. The end away from the cylinder sits on the ground when spitting vertically. If that end were the wedge, you'd be pushing the split pieces into the ground (or trying to lift your splitter off the ground).
A stationary block means the pieces stay right where they are when split (assuming you have a cradle or table to catch them), so they are right there if you need to resplit them or within easy reach if you are tossing the pieces somewhere after splitting. This advantage alone is enough for some people to choose a stationary block, even if they are never using a horizontal/vertical splitter in vertical mode. The downside is that if you are not tossing each piece, the split pieces rapidly fill up the area where you are standing and also surround the splitter making it harder to move if you want to relocate before clearing the pile. The falling pieces can also be a hazard to toes, the engine, or other parts depending on where those parts are located on the splitter. (A well designed splitter of the stationary block type will locate sensitive/expensive parts elsewhere or shield them.)
A stationary wedge pushes the split pieces off the end of the splitter, or onto an exit table if the splitter has one (highly recommended). The downside is that pieces which need to be resplit have to be pulled back to the splitting area (or picked up off the ground, if you do not have an exit table). The upsides are that the splitter pushes split pieces out of the way - even with an exit table, you can use the next log being split to push the previously split pieces off the end of the table. This puts the pile and the falling pieces off the end of the splitter and more out of the way. When the pile grows to where that area is full, it's usually possible to just push the splitter back a few feet, creating more space for the splits to drop. A stationary wedge lends itself more easily to do using a 4-way wedge, particularly one which is easily removeable. People do have 4 way wedges on moving wedge/stationary block splitters, but they are more rare. Some of this is that designing a 4 way to fit the moving wedge is a bit tougher. Another factor is that 4 way wedges are more likely to put a significant side load on the wedge, especially with unevenly cut or gnarly logs. Designing for handle that side load is a lot easier with a stationary wedge.
The high-end splitters with a stationary wedge offer an option for an adjustable height for where the center of the 4-way sits, allowing it to adjust for different diameter logs. (usually hydraulic, though I have seen one set up to operate from a foot pedal and mechanical linkage).
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My splitter is the stationary wedge design. I had a horizontal/vertical splitter, but found that I hated using it in vertical mode. I ran horizontal all the time. It was a little light-duty for my needs, so I shopped for a used horizontal splitter that had a working height just a bit below waist high, and had a hydraulic log lifter. Mine does not have the adjustable wedge height option, but since the 4-way is easily removable, I had a custom 4-way made with the cross wedge was at a different height. It turns out I use my custom one more, since I like the smaller split size it offers. I rarely use the one hat came with the splitter.
I've also used my splitter to push pieces right off the exit table and into the back of a trailer, saving some handling. (I do have to go back and shuffle some pieces to get the trailer really full, but the walls of my trailer are not all that tall, so I've only got to do that a couple times per trailer load till it's full, and it's easier than picking them up off the ground.) If I'm lucky, I can get the trailer about half full before I have to do that shuffling. I've seen videos of a guy who split a plastic culvert in half lengthwise, creating a trough which he would prop up on the back of his pickup with the far end up in the air near the middle of the bed. He could push logs up his "poor man's conveyor" and do a pretty god job of loading his truck.
I've discovered that I much prefer the ability to push the pieces off the end of the splitter, even if it does mean pulling back the resplits. The way my table is set up, that pull back is not difficult (though I have seen splitters with exit and work tables that are even better). Since I have an abundance of logs generated from my wildlife habitat and timber stand improvement work on my property, I'll often pick logs which fit the sweet spot of one of my 4 way wedges, and just run them through once and done. You also develop techniques to improve the work flow with various sized logs (as I'm sure people do with any splitter).
