The short answer is that you need to use a collect statement to make it work.
MATCH (p)<-[rel]-(v:myDistinctNode) WITH collect(v) AS nodes RETURN count(nodes)
This should return one.
I'm not a cypher expert, but I believe the reason it doesn't work is that the cypher result seems more like a table where in one row you have p, another row you have r, and the last row you have v. Even though v is a unique entity, there are still 80 rows that have v.
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