When I courted my husband he was living in Tokyo with his father, and, for the most part, the hotel rooms I stayed in were too small and claustrophobic to do any entertaining there.
We'd often hang out with his friends (one owns an art gallery which he lives above of so that offered a lot of space) - when we wanted 'alone' time, we'd go out and explore Yanaka together.
Coffee shops are popular, you don't need to order that many drinks and can hang around in them for hours.
Karaoke booths seem to offer that 'in between' ground you're enquiring about. As well as groups of friends, many couples hire them for a couple of hours - you can get drinks and snacks delivered to the door of the 'room'.
Korakuen (the amusement park in Northish Tokyo) is full of dating couples. As is Odaiba. There's a brewery in Tennozu Island - which serves as a restauarant and as a pub/izakaya - that's extremely popular with couples.
But for MrLapLap and myself, I guess our main focus was walking together (in and out of Tokyo), taking a bento box and picnicking in a park, and coffee shops.