It is quite possible both terms were used and are similar enough that one appears to be the source of the other.

Possibly, the lighting systems were developments of each other.
Apologies if I’m misunderstanding you here, but there isn’t such a thing as ‘a funnel lighting’ system.

Originally there were various lighting systems in use, including the pre-war Chance floodlighting abd Contact lighting.
New designs came along to standardise systems such as Drem which was known as A.F.L. MkI.
MkII introduced funnels, including outer, intermediate, and inner funnel arrangements to pick up from the outer circle of lights and feed (or funnel)aircraft directly to the flare path and runway.
MkIII had main funnels and fog funnels.
I’m unsure at what Mk AFL it changed but funnels gave way to cross bar lighting on the approach and better radio approach aids.

Just out of interest, the funnel lights also had a take-off use whereby they supplied horizon reference to a departing aircraft crew at night.

It’s a huge subject, which AP3236 covers well.