I wish to count only the nodes from the red node with name="this is the first" and not the nodes connected to the red node with the name="not this one"
Try this:
//Nodes connected to "this is the first"....
match (d:RED_NODE) where d.id = 1
CALL apoc.neighbors.athop(d, ">|<", 10) YIELD node
with count(node) as lvl10
RETURN lvl10
//Nodes connected to "not this one"
match (d:RED_NODE) where d.id = 2
CALL apoc.neighbors.athop(d, ">|<", 4) YIELD node
with count(node) as lvl4
RETURN lvl4