Neither the function $$f(x)=\frac{x^2+x-6}{x-2}$$ nor the function $g(x)=x+3$ has a discontinuity at $x=2.$ In the first case, it is because $f$ is not defined at $x=2,$ so is not (continuous or discontinuous) there. In the latter case, it is continuous there.
Now, consider the family of functions $h_\alpha(x)$ given by $$h_\alpha(x)=\begin{cases}x+3 & x\ne 2\\\alpha & x=2,\end{cases}$$ where $\alpha$ is a real constant. The function $h_5(x)$ is simply $g(x),$ expressed in a more complicated form. For any real $\alpha\ne5,$ $h_\alpha(x)$ has a removable discontinuity at $x=2.$ What this means is that we can make $h_\alpha(x)$ into a (different) function that is continuous at $x=2$ by choosing a different value for $h_\alpha(2).$
Consider the function $i(x)=\frac{x-2}{x-2},$ now. Note that $i(2)$ is undefined, and so $i(x)$ doesn't make sense unless $x\ne2.$ And, of course, for $x\ne2,$ we have $i(x)=1.$ Now, in order to multiply $g(x)$ by $i(x),$ both of these have to be defined, and so we have to require that $x\ne 2$ once again. So, while $g(x)$ is defined everywhere, $f(x)=g(x)\cdot i(x)$ is not.
Added: It's worth noting that the sources to which you linked in the comments are not using the standard definition. (However, this abuse of terminology is far from uncommon, even among textbooks!) Moreover, their definitions don't even agree with each other!
The first source (or, at least, what I can read of it) seems to adhere to an informal definition that suggests the following formal definition:
A function $f$ has a removable discontinuity at a point $x_0$ if there is a number $L$ such that $\lim_{x\searrow x_0}f(x)=L=\lim_{x\nearrow x_0}$ and either (i) $f(x_0)$ is undefined or (ii) $f(x_0)$ is defined and is not equal to $L.$ In the former case, we say that $f$ has a hole at $x_0.$ In the latter case, we say that $f$ has a created discontinuity at $x_0$.
The terminology "hole" (and this definition of it) is fairly standard. However, the definition of "created discontinuity" (a term I've never seen before) is exactly the standard definition of "removable discontinuity"!
The sketched graph provided there gives an example of a hole, and the source also seems to refer to a graph giving an example of a "created discontinuity," but it isn't visible to me.
The second source, on the other hand, conflates the terms "hole" and "removable discontinuity," as opposed to your first source (which treats holes as special cases of removable discontinuities) and the standard usages (in which holes and removable discontinuities are mutually exclusive). Moreover, its attempt at a formal definition is at best informal, and at worst self-contradictory!
Formally, a removable discontinuity is one at which the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point.
The part preceding the semicolon is the usual definition, as, by speaking of "the value of the function at that point," it is implied that the function takes a unique value at that point! However, this is directly contradicted by the part after the semicolon. Thus, if we attempt to make it formal, we fail. If we leave it informal, we can ignore the implication and thereby avoid the contradiction (but not potential confusion).
The graph provided there gives an example of a removable discontinuity (as usually defined), but not of a hole (as usually defined).