For the case of a closed oriented surface $S$, its first Betti number $\beta_1$ and its genus $g$ are related by the equation
$$\beta_1 = 2g
$$
Those two quoted phrases therefore cannot be the same. Notice that the cut procedure in the phrase defining genus is quite precise, whereas the cut procedure defining Betti number is rather vague, and therein lies the difference. You might want to follow up the reference given in the Betti number definition, which occurs right after your quote cuts off. One traditional way to describe $\beta_1$ is as the maximal number $2k$ of simple closed curves in $S$ that can be listed as $a_1, b_1, a_2, b_2,...,a_k,b_k$ so that if $i \ne j$ then $a_i$ and $b_i$ are disjoint from $a_j$ and $b_j$, and so that $a_i,b_i$ intersect each other transversely exactly one time.
Your question also raises several other issues, and I'll treat these briefly.
The equation $\beta_1 = 2g$ that relates the first betti number and the genus can be deduced by comparing the actual definition of $\beta_1$, namely the rank of the first homology group, with the actual calculation of the first homology group of the surface (carried out by using any of the calculational procedures learned in algebraic topology). I don't think it's particularly fruitful to over-interpret the meaning of an intuitive explanation of betti number in terms of "number of holes". The purpose of that intuition is to get you started, and you could keep reading other posts on that issue if you desire, but really you should dig into the technical details of homology groups to learn what's really going on.
In particular, the relation between the Euler characteristic and the genus can again be deduced by working through the calculations of homology groups.
If you want more detail, it would be better to ask separate and more precisely worded questions, instead of lumping too many vaguely worded questions into one post.
Best Answer
"Cutting a surface into two pieces" corresponds to the case of a $2$-manifold (= surface) and $n=1$, in more generality it's the rest of the sentence: "two pieces or 0-cycles, 1-cycles, etc."
In the case of a surface and $n=1$, the first Betti number $b_1$ does correspond to how many "cuts" you can make before getting more than one connected components.
Keep in mind that this is only an informal explanation, and with increasing $n$ it's harder and harder to understand what $b_n$ "means" (how many times you can cut your space with an $n$-cycle and still keep one $(n-1)$-"piece").