[puts on puzzle wonk hat]
First, all these discussions of where to put puzzles in a hierarchy strike me as failing by using sudoku as a touchstone. While this is a convenient choice as sudoku is one of the most recognized (and misrecognized) logic puzzles these days - sudoku seems synonymous with puzzle in some circles - it is well zoomed in on the broad puzzle landscape already. So the fact almost no other puzzle qualifies as a sudoku, but many puzzles can be turned into sudoku forms, seems missing here when the discussion turns to whether a thing (say Kropki) is a sudoku or not.
In the broad hierarchy of puzzles, there are puzzles that should be classified as symbol placement puzzles (as opposed to things like path/loop-forming puzzles, a similarly broad category that would get mazes and Slitherlink and Numberlink and such, or observational puzzles or manipulative puzzles and perhaps a few other groups). Within symbol placement puzzles are puzzles that literally just use symbols (like Star Battle) but also more meaningful symbols such as word and number-based subgenres. Some of those puzzles would be things like word fill-ins and Fillomino that have different types of logical rules but no specific geometric uniqueness constraints. There are also symbol placement puzzles that have these unique group constraints, of which Latin square puzzles are the most common. Within this family, Kropki, Calcu-doku, Skyscrapers, even Easy as ABC puzzles belong. That some take numbers and others letters is a problem if we are overly strict in how we classify things. A subset of Latin square puzzles, with a third unique group constraint based on regions, are Sudoku. Without the third constraint - specifically a region-based third constraint - you do not have a sudoku in my opinion. A calcu-doku has a third constraint (based on sums/products/... of cages), but this is not a region-based constraint and therefore not a sudoku. It feels close, because it is close, but it is not an exact cousin of a sudoku.
I feel there is a job for some Linnaeus of puzzles to formalize this a bit more with kingdom and phylum all the way down to genus and species, but I'd call Sudoku a "Symbol-placement number-based puzzle with three-way uniqueness constraints on each cell". A typical Kropki puzzle is a "Symbol-placement number-based puzzle with two-way (Latin square) uniqueness constraints on each cell as well as numeric property-based constraints between pairs of cells". A Kropki sudoku merely changes two-way to three-way in the above, but until it does so it is not a sudoku. Certainly more closely related as a puzzle to a sudoku than a spot the differences to a sudoku as it belongs in the symbol-placement kingdom, number-based phylum, and similar uniqueness-group class, but no more a sudoku than a chimp is a human for all its genetic similarities.
That's the last I think I'll write here about this. It seems each new entry in the sudoku forum re-inspires this debate until the topic ends up being moved to "puzzle". But that seems the correct motion in every case. Unless you do mean "symbol placement" or "number-based" or "unique element constraints" to always stand for sudoku. Then just be clear about that fact.
Second, and more interesting conversationally, with the concept of trademarking/patenting a puzzle, certainly someone can protect the trade name of their product. Within Japan, Sudoku cannot be used by anyone other than Nikoli. Nikoli did not seek world-wide protection on this name though, so it is free to use outside of Japan by any publisher. Everyone, from Shortz to me, can call their puzzles Sudoku. No one else calls their's "wordless crossword puzzles" - and this does not help the puzzle hierarchy section above if sudoku is suddenly a subtype of crossword puzzles - but I could imagine St. Martin's could have some trade protection on that marketing terminology which is strictly that - marketing terminology.
Nextoy has taken the steps to trademark KENKEN and KenDoku and are the official suppliers of that name/brand of puzzle, but not that type of puzzle. They do not and cannot claim exclusive rights on Latin square puzzles involving arithmetic. First off, they would have a heck of a time proving they had "invented" the concept in the mid 2000s considering decades of prior published puzzles. If such minor changes count as new ideas, then my using multi-cell subtraction already would qualify as a further "new" design and wouldn't be a problem anyway. Second, as mentioned up-thread, "concepts" are harder to protect than tangible objects. So individual puzzles/books cannot be copied, but derivative (or improved) work in the same style is allowed. Hence, things like TomTom Puzzles.
A gray area exists though when it comes to puzzle presentation itself. Last year, after a breakdown of a relationship between the Grabarchuk family and Conceptis puzzles, the product Strimko was released as Chain Sudoku with identical presentation (sets of circles, lines, and colors) which extend beyond the rules of the puzzle itself. Here, I think legally Conceptis is in the clear, but morally they ripped off another product and design to release their own version, particularly considering the state of discussions that I'm aware of between the parties before this release. Look for the story to become the basis of The Social Network 2 if it ever becomes a billion dollar product.
So, when I write puzzles - realizing I am often making new types and borrowing other's types, I do my best to give credit when I am taking an idea and don't mind when others take styles I have developed and use them for their own designs. I do care when you steal my puzzles, or reprint them without using my name. Admittedly some of my work is now in the public domain, such as WSC example puzzles, but still when an authorship or credit can be provided it is worth giving that credit if known. And hoping for anything more than credit when an idea is used or stolen by someone else is simply not reasonable given the ubiquity of new puzzle evolution in public locations.
[/takes off puzzle wonk hat]