The conventional wisdom is to create a minimum viable product (MVP), release it, get customer feedback, use that feedback to improve the product, release again, get more feedback and so on.

In terms of releasing to a closed user group, that is fine but when you launch a new website you're probably doing that anyway in that no one will have heard of it unless you through lots of money at online marketing or get lots of PR.

In other words, usually it will take a while for your website/web app to get lots of traffic so a public MVP should not be an issue for reputation etc.

On the subject of MVP, it should be the minimal amount of time, effort and money needed to develop the product to a working point, the most basic version of the site. Of course if s/he is just re-skinning something else, I'm not sure that is an issue.