My textbook says
Definition 1: A graph, G, is acyclic if it contains no undirected cycles (otherwise it’s cyclic).
It also says
Definition 2: A (directed) cycle is a (directed) path which begins and ends at the same vertex. An undirected cycle is, likewise, a path beginning and ending at the same vertex which may or may not respect edge directions.
These definitions confuse me.
By Definition 1, can a graph be acyclic and yet contain a directed cycle? This sounds like a contradiction, but the definition only says an acyclic graph should not contain undirected cycles and says nothing about directed cycles. Unless Definition 1 is implying that all directed cycles can be treated as undirected cycles, but undirected cycles cannot be treated as directed cycles?
Definition 2 seems to reinforce this idea, by suggesting that an undirected cycle can simply ignore edge directions.
On the other hand, I found this website which claims this is a directed acyclic graph:
But by Definition 2, I can just ignore edge direction and create undirected cycles, like $1\rightarrow 2 \rightarrow 5 \rightarrow 6 \rightarrow 3 \rightarrow 1 $, which would make the graph cyclic because it contains undirected cycles. However, the website says it is acyclic, which contradicts everything I've said.
I think I am probably just misinterpreting all of these definitions. Can we define the terms "acyclic", "cyclic", "undirected cycle" and "directed cycle" in some other way to help clarify what they mean?