No there is no specific name for the people of God but in the beginning, it was easy to tell who they were because no one was like them. Eventually, as we see from the scriptures, more and more came in and were false and things progressively got worse.

If it were not important to identify each other we would not go by anything. But we have bible pattern shown that they went by Christian publicly but among themselves they called themselves various things.

We do the same today when talking to each other. Today we also face melting into the mass of falsehood if we do nothing so we can find each other.

These are difficult and trying times we live in today. We must walk with real caution with respect to not only autonomy but to the church universal in Christ, since we are all in Christ and must consider how we as congregations of the body are joined into the body world wide. Although the hand is separate from the ear, they still are attached into one body.

1 Corinthians 12

12 For as the body is one and has many members, but all the members of that one body, being many, are one body, so also is Christ. 13 For by one Spirit we were all baptized into one body—whether Jews or Greeks, whether slaves or free—and have all been made to drink into[c] one Spirit. 14 For in fact the body is not one member but many.
15 If the foot should say, “Because I am not a hand, I am not of the body,” is it therefore not of the body? 16 And if the ear should say, “Because I am not an eye, I am not of the body,” is it therefore not of the body? 17 If the whole body were an eye, where would be the hearing? If the whole were hearing, where would be the smelling? 18 But now God has set the members, each one of them, in the body just as He pleased. 19 And if they were all one member, where would the body be?
20 But now indeed there are many members, yet one body. 21 And the eye cannot say to the hand, “I have no need of you”; nor again the head to the feet, “I have no need of you.” 22 No, much rather, those members of the body which seem to be weaker are necessary. 23 And those members of the body which we think to be less honorable, on these we bestow greater honor; and our unpresentable parts have greater modesty, 24 but our presentable parts have no need. But God composed the body, having given greater honor to that part which lacks it, 25 that there should be no schism in the body, but that the members should have the same care for one another. 26 And if one member suffers, all the members suffer with it; or if one member is honored, all the members rejoice with it.
27 Now you are the body of Christ, and members individually.


Acts 11:26
And when he had found him, he brought him to Antioch. So it was that for a whole year they assembled with the church and taught a great many people. And the disciples were first called Christians in Antioch.