That's where the No True Scotsman fallacy comes in. There are obvious criteria for being certain things. For some things the problem comes in defining the absolute core requirements of membership that nobody can disagree with. You can say he's No True Scotsman if he's from Bulgaria. It's easy with the Scotsman, where was he born? Christianity is more difficult, but I think the three requirements you mentioned are the core ones, and anything beyond that represents your particular interpretation of what it means to be Christian, thus invoking the fallacy.
Christ said you will know them by their fruits and he was speaking about going out and sowing the word so God would reap a harvest.
Mr. Phelps puts more effort into sowing the word of God as he sees it than most Christians, that's why he's famous. Personally, I think he just puts too little emphasis on the New Testament, instead emphasizing the pre-covenant angry God.
The Gospel is completed with the new testiment you can not have a gospel without it..