Definition of a TM accepting a word
There are two reasons to use this kind of definition:

It is a formal way to define the run of the machine.
It generalizes to the nondeterministic setting.

In the deterministic setting, an equivalent definition would be:

Define a (possibly finite) sequence $C_1,C_2,\ldots$ as follows:

$C_1$ is the start configuration of $M$ on input $w$,
For $i \geq 1$, $C_{i+1}$ is the unique configuration yielded by $C_i$, if such a configuration exists.

The machine $M$ accepts the input $w$ if the sequence is finite, and the last configuration is accepting.

This is not dramatically better than the quoted definition, and it doesn't generalize to the nondeterministic setting. It is a matter of taste which of the two to prefer.