Tyler James Young's comment really should be an answer - or it should be the answer :))
Common usage implies singular in both instances. But there are differences that can be subtle. Always consider the simple and most normal usage first.
So recall those grammar and usage lessons: "a" and "an" are the indefinite articles. "The" is the definite. The former can be general, the latter specific. Though this is generally true, the context can change this.
- I ate an apple. [any apple]
- He ate the apple. [ Hey! That one was my apple, buddy ! ]
- Adam ate the apple Eve gave him. [ Really The only one like ever like that :) ]
So how to apply this to the houses in Oxford?
It reminded me of ...
the house I rented in Oxford.
The only one you rented there. (And, obviously, there's more than one house in that location.)
a house I rented ...
You probably rented one only, but there are others you might have chosen. It's less specific.
But if you said something like:
It reminded me of a red, very small house with a pointy tile roof I rented ...
Then your friend might have replied:
Oh, you rented the Farthingworth's house ?
(No, there's probably no such place - it's just an illustration :))
She might have also said:
Oh, you rented a Farthingworth house ?
Because the Farthingworths are quirky Oxford landowners - they paint all their houses red and put pointy roofs on them. (Note also the implied possessive "Farthingworth house".)