[Math] what is Path and how to find number of possible paths of length $r$

discrete mathematicsgraph theoryterminology

The word path is used in different way in different contexts. But I can't related them with each other. Like

geeksforgeeks ${}^1$Path: It is a trail in which neither vertices nor edges are repeated i.e. if we traverse a graph such that we do not repeat a vertex and nor we repeat an edge.

here Path: A path in a graph is a subgraph of a given graph that is isomorphic to a path graph.

kenneth rosen discrete mathematics Path: Let n be a nonnegative integer and G a directed graph. A path of length n from u to v in G is a
sequence of edges $e_1,\cdots,e_n$ of G such that $e_1$ is associated with $(x_0,x_1)$, $e_2$ is associated with $(x_1,x_2)$, and so on, with $e_n$ is associated with $(x_{n-1},x_n)$, where $x_0=u$ and $x_n=v$. When there are no multiple edges in the directed graph, this path is denoted by its vertex sequence
$x_0,x_1,x_2,\cdots,x_n$. path of length greater than zero that begins and ends at the same vertex is called a circuit or cycle. A path or circuit is called simple if it does not contain the same edge more than once.

Question $2$: How many paths of length four are there from $a$ to $d$ in the simple graph $G$ in Figure $8?$enter image description here
So there are exactly eight paths of length four from a to d according to their solution. But it can't possible without $\color{red}{\text{repeated vertices}}$. Which seem conflicted to the first definition${}^1$. So I really confused which definition should I follow or I have misunderstood something with these defintions. Besides provide me a list something like:

$(1)$ Walk can be repeated anything
$(2)$ Vertex can be repeated Edges not repeated in Trail
$ \vdots $

Thanks for your time.
Thanks in advance .

Best Answer

See my answer here and Matthew Daly's comment. In this case, I would say (and many others) that the $ij$th entry $a_{ij}$ of $A^n$ is the number of walks of length $a_{ij}$ from $v_i$ to $v_j$, where $A$ is the adjacency matrix (More information in this stack exchange question.)

In this particular text, it looks like path is defined to be as what is otherwise referred to as a walk, and then they use simple to clarify, which is not uncommon.

I'd recommend taking a look at Douglas West's Introduction to Graph Theory if you can, specifically sections $1.1$ and $1.2$. I believe West defines a path graph in section $1.1$ and implicitly uses the "isomorphic to a path graph definition" in section $1.2$.