How to make an "invisible" edit in Wikipedia?

Not possible to make invisible edit in Wikipedia, because all content reviewed by Wikipedia, if it does not according to it policy, they warn you and can block. 

What's the reason that you would want to make an "invisible" edit?

It is not possible to make an invisible edit to an article in Wikipedia. Every article in mainspace has a history tab, right there with the article itself and the talk page. Even if you make a change to an article, then change it back to exactly what it was before, there will be an audit trail. 

For every edit to a Wikipedia article, the history tab will record - your user name (or IP address if you don't have or don't choose to use a Wikipedia editor user name),- the year, month, day, and time you made the change to the article,

- the number of alphanumeric characters you added to or removed from the article.

If other editors make changes after you, it is easy to see exactly what words, photos, etc. they added or removed from the article. Even Wikipedia editors who are admins (or have other high level privileges) cannot hide changes they make to an article.

such an animal does not exist...

It's amazing how wishful thinking and naivety can so easily devolve into pure stupidity, isn't it?

Ask on wikipedia. There are many excellent resources there.