Catalyst verses Reaction Intermediate? Please help me to understand?

Catalysts are supposed to be used in the reaction to speed it up via a different mechanism, and they would get produced again at the end of the reaction cycle.

Intermediates are supposed to be made in the middle of the reaction and then get used up by the time the reaction is over.

In the mechanism, we have:

#cancel("NO"_2(g)) + "SO"_2(g) -> cancel("NO"(g)) + "SO"_3(g)#
#cancel(2) "NO"(g) + "O"_2(g) -> cancel(2) "NO"_2(g)#

It adds up to an overall reaction.

#"NO"(g) + "O"_2(g) + "SO"_2(g) -> "NO"_2(g) + "SO"_3(g)#

This is not a great example, but apparently,

• one of the #"NO"# molecules is a reactant, and one of them is an intermediate.
• one of the #"NO"_2# molecules is a product, and one of them is a catalyst.

Here's a better example.

#color(green)(cancel("Cl"(g))) + "O"_3(g) -> color(purple)(cancel("ClO"(g))) + "O"_2(g)#

#ul(color(purple)(cancel("ClO"(g))) + "O"_3(g) -> color(green)(cancel("Cl"(g))) + 2"O"_2(g))#

#2"O"_3(g) -> 3"O"_2(g)#

Here, the #color(green)("catalyst")# is #"Cl"(g)#, and the #color(purple)("intermediate")# is #"ClO"(g)#. Why is that?