Good question. Fascinating NGram supporting Kate's comment about back in the day being a recent development.
Google Ngram Back in The Day vs Back in the Days
To some extent your sentences are not interchangeable. Your second sentence should read something like -
Back in the days of paying rent, we had an apartment with a swimming pool.
Which is like saying -
Back in the day, when we were renters, we had an apartment with a swimming pool.
Back in the Day, is a slangy way of saying, when I was younger or some time ago. The phrase is usually followed by a anecdote describing what someone experienced or how they behaved in their past.
Back in the day, we had to change TV Channels by turning a tiny knob on the TV.
Back in the days is the more traditional usage of the phrase and its usage is slightly different. It is used to establish a time frame and is intended to be more specific than its slangy counterpart. It is generally followed by some illustrative phrase, often starting with the word 'of', intended to establish a time period.
Back in the days of black and white television, we changed channels by turning a tiny knob.