Quoted from the text (bolded by me):
The word does not have quite such a full-blooded meaning now as it had in the times of St. Augustine and St. Thomas Aquinas. In those days, if a man said that he was a Christian it was known what he meant. You accepted a whole collection of creeds which were set out with great precision, and every single syllable of those creeds you believed with the whole strength of your convictions.
What Is a Christian?
Nowadays it is not quite that. We have to be a little more vague in our meaning of Christianity. I think, however, that there are two different items which are quite essential to anybody calling himself a Christian. The first is one of a dogmatic nature -- namely, that you must believe in God and immortality. If you do not believe in those two things, I do not think that you can properly call yourself a Christian. Then, further than that, as the name implies, you must have some kind of belief about Christ. The Mohammedans, for instance, also believe in God and in immortality, and yet they would not call themselves Christians. I think you must have at the very lowest the belief that Christ was, if not divine, at least the best and wisest of men. If you are not going to believe that much about Christ, I do not think you have any right to call yourself a Christian.
I think it is quite clear from the text that Russel refers to the Scholastic tradition where Christians were pretty well defined and there had been no pluralism whatsoever blurring the definition. There has been the catholic church and some specific orders within it. That's about it. So nowadays, the definition has to be more vague in order to be applyable to all varieties that are thought under the concept of "Christianity". Think about all the sects, religious communities, churches etc. that are considered "Christian", not to mention evangelists and the variety under that name alone.
One example illustrating this is Eucharist (quotes from Wikipedia):
The Catholic Church teaches that once consecrated in the Eucharist, the elements cease to be bread and wine and actually become the body and blood of Christ, each of which is accompanied by the other and by Christ's soul and divinity. The empirical appearance and physical properties are not changed, but for Catholics, the reality is.
In the Reformed Churches the Eucharist is variously administered. The Calvinist view of the Sacrament sees a real presence of Christ in the supper which differs both from the objective ontological presence of the Catholic view, and from the real absence of Christ and the mental recollection of the memorialism of the Zwinglians and their successors.
And this is only one of the many differences presented there.
To take another example out of another religion: Islam for sure had been one canonical religion at one point. This would have been the situation similar to the one Russel refers to when speaking of St. Thomas for Christianity. But look at the variety presented here, some of them do not even consider the other to be true muslims. And I think there will be catholics that do not consider calvinists to be true christians. Nevertheless, they all are muslims respectively christians and the definitions must be applyable for all the varieties.