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The Nov 2019 White Dwarf Inquisition rules don't mention anything about reinforcement costs for the demon host Eisenhorn summons with his Malus Codicium ability. This is a departure from Eisenhorn's old rules which explicitly called out the reinforcement points cost of using the ability at the end of the Malus Codicium description. Does this mean Cherubael is now included with Eisenhorn's 80 points?

If the demonhost needs to be paid for separately, can it be listed as an Elite in the detachment? Does it have to be listed?

There is no need to mention any reinforcement points. Any new unit added always costs reinforcement points, unless the rule itself otherwise.

BRB update 1.0 (released over 24 month ago !)

REINFORCEMENTS
Q: If a rule creates a new unit during the battle in a matched
play game and adds them to my army, must I pay for the unit
with my reinforcement points?
A: Yes (unless the rule itself says otherwise). If you don’t
have enough reinforcement points, you cannot add that
unit to your army.

A summoned unit isnt part of any detachment, it cant be on your army roster. Instead you write how many reinforcement points you have and use those to summon units. If you set enough points aside you can summon a new cherubael if the old one dies.

Awesome, thank you!