Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{P}^n$ by $G$ is not isomorphic to $\mathbb{P}^n$ but yet is locally a set-theoretic complete intersection (if we embed it into $\mathbb{P}^m$ for some $m>n$)? It seems that the most "popular" quotients of this type are the so-called weighted projective spaces; so what is known for them? Can $P$ be smooth, and still not isomorphic to $\mathbb{P}^n$?

I am also interested in cases when $P$ can locally be "described by not so many extra equations" (say, by $m-3/4 n$ for $P\subset \mathbb{P}^m$) and when the "non-LSTCI locus" (i.e., the set of points where $m-n$ equations are not sufficient) is of dimension less than $n/4$.

Upd. I am deeply grateful to Francesco Polizzi for his answer (including examples). So I would like to ask the following: can one describe weighted projective spaces (or more general $P$ as above) that are locally (set-theoretic) complete interestions? Is it possible to calculate the number of "extra local equations" for weighted projective spaces? Any references (and electronic versions of these text) would also be very welcome!