Definition of a bounded sequence

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...

Both your Prof. and wikipedia is right infact your Prof. is more precisely telling you where that $M$ lies.

The only difference between both the definitions is that your Prof. take $M\gt 0$ and wikipedia is taking $M\in \mathbb R$ but don't you think wikipedia definition also indirectly says that $M\ge 0$ because $|x_n| \ge0 ,$thus it can't be made smaller than a negative number. And $M=0 \Rightarrow |x_n| \le 0 \Rightarrow x_n=0$ and if sequence is constant zero then any $M\gt 0$ will also work.