Call a coinitial collection of arrows $\{f_i:X\to Y_i\}_{I\in I}\subseteq{\bf Hom}_\mathcal{C}$ in a category $\mathcal{C}$ an $I$-indexed source. For a category $\mathcal{C}$, call an ${\bf Ob}_\mathcal{C}$-indexed source a $\mathcal{C}$-indexed source.
Using this language, a cone to a functor $F:\mathcal{C}\to\mathcal{D}$ is just an $\mathcal{C}$-indexed source in $\mathcal{D}$ with $Y_i=F(i)$ for all $I\in{\bf Ob}_\mathcal{C}$ and a coherence condition relating members of the source in the 'natural' way via composition with members of $F({\bf Hom}_\mathcal{C})$.
Definition. Let $F:\mathcal{C}\to\mathcal{D}$ be a functor, with $S:\mathcal{I}\to\mathcal{C}$ a diagram of shape $\mathcal{I}$ in $\mathcal{C}$. We say that $F$ creates the limit of $S$ iff $F\circ S$ having a limit $$\pi':\Delta L'\Rightarrow F\circ S$$ in $\mathcal{D}$ implies that there exists a unique (up to iso) $\mathcal{I}$-indexed source $$\hat L=\{\pi_I:L\to S(I)\}_{I\in{\bf Ob}_\mathcal{I}}$$ in $\mathcal{C}$ such that $$F(\hat L)\cong\pi'$$ as cones, and further that $\pi:\Delta L\Rightarrow S$ is a limit of $S$ in $\mathcal{C}$. We say that $F$ creates limits of shape $\mathcal{I}$ iff $F$ creates the limt of all functors $S:\mathcal{I}\to\mathcal{C}$, and that $F$ creates limits iff $F$ creates limits of shape $\mathcal{I}$ for all 'small' categories $\mathcal{I}$.
This definition is very similar to the one found in The Joy of Cats by Adamek, Herrlich, and Strecker (p. 227 definition 13.17), but they required uniqueness on the nose for $\hat L$ and that $F(\hat L)=\pi'$.
The discussion and lemmas in their book after their definition establish that it implies all other kinds of 'things you want a functor to do to limits' (c.f. the diagram in remark 13.38, p. 232, reproduced below), with the exception of preservation.
The definition above also implies all the good behavior one could ask for (except preservation), and is 'respected by equivalence' better than the one in JoC since it makes no reference to uniqueness on the nose or equality -- equivalences create limits in both senses of the definition though, and I would argue that both are better than the definitions in CWM, the nlab and the notes you reference.
The subtle difference between the JoC definition and most other definitions encountered is that we don't a-priori impose the cone coherence conditions on the source in the domain when we assert that it is unique; it must be unique as a source, not as a cone. We then 'create' the cone coherence conditions using $F$; in the case of equivalences this is very straightforward to do using essential surjectivity and full faithfulness.
EDIT: Since this definition doesn't imply preservation of limits, if preservation of limits matches our intuition for what 'creating a limit' should mean we ought add that $F$ preserves limits to the above definition.