Well, if that were indeed a sufficient definition, everything that we find said about the institutional Church in the New Testament wouldn't mean much. At the least, all of that would be optional.
So, that in turn would mean Christ commissioning the first bishops, the Apostles, instructing them to evangelize and baptize, plus his directive to observe the Lord's Supper and about its importance, etc. etc. would not, in fact, be particularly important. Or maybe we'd say in that case that all of this applied ONLY to the Twelve Apostles. This certainly would not be something every believer SHOULD do.
While Christianity has had its share of hermits, etc. over the centuries, I cannot quite buy into a "take it or leave it as you wish" posture regarding this matter, and neither has the Church throughout the ages done so. Incidentally, that's why the early hermits were gradually gathered into monastic communities.
Therefore, if we say that "Christianity consists of Christians who trust in the saving work of Jesus Christ...." we are actually describing Christians rather than Christianity.