Personally, I think the supposed established Japanese characterization of the term 'harem' tippertop presented was rather unconvincing.
I'd be very curious to dig around old usenet archives to see when the term 'harem' first started being used to denote stories of a man being romantically involved with multiple women, but most of those posts have been lost IIRC. I'm asking because the Tenchi Muyo manga was first published in 1992, the anime aired in Japan in 1995, and in America in 1997, so the term should have started to be used in such a fashion in the English speaking sphere around that time as well.
To me that sounds more like open hunting season than a harem.
The majority of the plot usually involves the women fighting over the protagonist and him getting caught in the crossfire. It's mostly played for laughs, though there are titles that deal with the issue seriously and show the inherent tragedy of such a state of affairs.
That can't be the question. It only can be a harem if he lives together with multiple women, otherwise, it's not.
Yes and no. Tenchi Muyo was the first story to introduce the concept of the male protagonist cohabiting with multiple women that were romantically interested in him, and Love Hina, and the rest of the newly minted harem genre followed along.
A more modern interpretation of the genre uses 'harem' to denote that the story features a gender imbalance among the cast, usually with far more female characters than male ones (to the point that sometimes the protagonist is the only male around) but does necessarily imply that the women are in love with the male lead, though this is a much rarer occurrence.
One thing that is universally prevalent in the genre though is that there is no "harem poaching", at least not until the protagonist has made his choice. In fact, authors that have tried to broach the subject have been met with rather heated hostility by audiences.
That's why I object to the use of the 'harem' tag when there is sharing of partners going on. Neither historically, nor in the context of internet use, does 'harem' denote stories where the women sleep around with other men. And like others have said, there is really no need to use the 'harem' tag in such an event because there are already other tags that can be used for such a purpose.