It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and write some errata emails.
A standard definition of a limit of a diagram is that it’s a terminal object in the category of cones to the diagram. That is, a limit of a diagram is a cone to the diagram with apex $C$ such that for any cone to the diagram with apex $C'$ there is a unique morphism from $C'$ to $C$ that makes everything commute.
I was surprised to find that quite a number of sources replace “any cone” by “any other cone”. Wikipedia:
A limit of the diagram $F:J\to C$ is a cone $(L, \phi)$ to $F$ such that for every other cone $(N, \psi)$ to $F$ there exists a unique morphism $u:N\to L$ such that $\phi_X\circ u=\psi_X$ for all $X$ in $J$.
The definition has included “other” since this edit $16$ years ago. Similar formulations are in A First Course in Category Theory by Ana Agore (p. $110$, “for any other cone”) and nLab (“every other cone”). The notes Introduction to category theory by Valdis Laan, while using “every cone” in Definition $5.42$ of a limit on p. $33$, have “any other object” in Definition $5.1$ of a product on p. $24$ and “any other morphism” in Definition $5.23$ of an equalizer on p. $30$. Category Theory for Programmers by Bartosz Milewski even revises its own definition of a terminal object on p. $56$ (“The terminal object is the object with one and only one morphism coming to it from any object in the category.”) in referring to that concept in the definition of a limit on p. $198$ (“Now we can define the universal cone as the terminal object in the category of cones. The definition of the terminal object states that there is a unique morphism from any other object to that object. In our case it means that there is a unique factorizing morphism from the apex of any other cone to the apex of the universal cone. We call this universal cone the limit of the diagram”).
By contrast, the majority of sources impose the uniqueness requirement for all cones, including the limit itself, for instance Basic Category Theory by Tom Leinster (p. $119$) (the best introduction to category theory that I’ve found so far) as well as the classic texts The Joy of Cats by Adámek, Herrlich and Strecker (p. $194$) and Categories for the Working Mathematician by Saunders Mac Lane ($2^{\text{nd}}$ edition, p. $68$).
My first reaction was that while the formulation with “other” seems unnecessarily complicated, perhaps it doesn’t matter because uniqueness for the limit itself is implied by uniqueness for all other cones. But that doesn’t seem to be the case.
Consider this category, with $e\ne1_C$, $e\circ e=e$, $f\circ e=f$, $g\circ e=g$:
The categorical product $A\times B$ is the limit of the diagram that maps the two objects of the discrete category $\mathbf 2$ to $A$ and $B$. Clearly the only cone to that diagram is the one with apex $C$ and morphisms $f$ and $g$. So if we apply the “any other cone” definition, then vacuously, since there’s no other cone, $C$ would be the limit of that diagram, and thus the product $A\times B$. But according to the conventional definition it’s not the limit and thus not the product because there are two different morphisms, $e$ and $1_C$, that make this diagram commute:
Note that in this example the limit is still unique even under the definition that includes “other” (since there’s only one cone). I don’t know whether this still holds in general, but certainly the standard proof of the uniqueness of limits up to isomorphism no longer works because it makes use of the uniqueness of the universal morphism from the limit to itself. None of the sources with “other” in the definition seems to provide an alternative proof of this fundamental fact; for instance, Ana Agore on p. $112$ infers the uniqueness of limits from the uniqueness of final objects which is proved on p. $11$ by applying the uniqueness of the morphisms to final objects to the final object itself. So it doesn’t seem that these texts are deliberately using a slightly different definition than everyone else.
So am I missing something here, or did all these people get this detail wrong?
I agree with you and both commenters that the word "other" doesn't belong here. In particular, this would give two inconsistent definitions of "terminal object" itself, since of course a terminal object is the limit of an empty diagram. The subtly wrong definitions that you quote would make the unique object of any one-object category terminal, for instance, which is definitely not what we want.
I certainly don't think these authors are intending the mathematically unnatural "for any cone not equal to the given cone" when they say "other", but just writing in more characteristic English and becoming a bit imprecise in doing so. Actually, it's an interesting general question about mathematical terminology--if $x$ is in $X$ and you say $\varphi(x)$ holds if for any other $y\in X,\psi(x,y)$ holds, I'd guess you almost certainly mean $\forall y\in X,$ not $\forall y\in X\setminus \{x\}.$ It'd be interesting to see examples from other areas.