Here's the situation:
Any ideas?
There may be different ways to derive points that represent clusters. Here is one idea (requiring Advanced/ArcInfo license):
1. Run Aggregate Points tool using 10m aggregation distance. You will get aggregated polygons.
2. Run Feature To Point tool on the polygons; you can choose either the Centroid or the Inside option to obtain a point per polygon. Are the resulting points good enough to represent their clusters?