In a monthly game night, my friend interprets the rules to say that if I have a village or city on my road, it reduces the "longest road" length. I interpret the rules as saying that my village/city on my road does not impact the road length.
Which is the correct interpretation?
You are correct.
Your own cities do not affect your status towards the "Longest Road" You can view the official rules here, from pg. 9 (emphasis is mine).
You can break an opponent’s road by building a settlement on an open intersection along his road!
On page 8, in the example, Emily has the longest road card with 7 segments. She has two houses within those 7 segments.
This was also answered directly in the online FAQ as pointed out by @shemmon.
To affect the road length, the village/city must be from another player.
And it only limits the counting of the road through that village/city. You can count up to the village/city that is not yours.
Also, here's a more official 'no:' https://www.catan.com/faq/4607-longest-road-do-my-own-settlements-interrupt-my-continuous-road
