Definition of a bounded sequence

My professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$

But then what about the sequence $(0, 0, ...)$? In that case, can't $M$ be $0$?

Wikipedia provides the following definition, which seems more reasonable to me:
A sequence $\{x_n \}$ is said to be bounded if $\exists M \in \mathbb R$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$

Is my professor's definition inaccurate or imprecise in any way?

The definition of your teacher is right. And the one from the Wikipedia is right, too. They are equivalent.

It is true that for the sequence $(0,0,\ldots)$ we have $|x_n|\le 0$ for every $n\in \Bbb N$, but this does not contradict your teacher's definition, since it says that a sequence is bounded if there exists some $M>0$ such that $|x_n|<M$.

In other words, your teacher's definition does not say that a sequence is bounded if every bound is positive, but if it has a positive bound. The sequence $(0,0,\ldots)$ has indeed a positive bound: $1$, for example (in fact, every positive real number is a bound for this sequence!)

No,it's fine. If you have the zero sequence $\{a_n\}$ then for every $M>0$ you have $a_n\leq M$. We define $M>0$ so we can use it sometimes to a fraction like ,let $\epsilon =\frac {1}{M}$. etc...