...the moon is always visible at night, and only at night,
A stable orbit that does this is impossible (there are however some compromises that are possible outlined below). Someone, somewhere on the planet will see the moon during the day. The 2nd lagrangian point (L2) is indeed the orbit that would place a planet in constant conjunction (i.e. located between) with another planet:
Image taken from wikipedia. The orbit that follows the path at L2 would indeed keep the interior planet between it and the star. However, the orbit is unstable: any perturbation away from L2 will continue to grow in time, causing eventual misalignment. If you are willing to ignore this fact, then it's mission accomplished. But if you need something with a bit more verisimilitude, I have a few potential candidates:
1. The geostationary Moon
Earth's moon currently orbits at a distance of $3.8\times 10^8$m and $27.3$ days. We could get the orbital period down to a single Earth day if it orbited closer at $4.2\times 10^7$ m, or about $1/9$ the distance. The advantage of doing this would be the moon would always be in the same place in the sky, day or night. It would travel through its full phase (new moon to full moon) in a single day. And it would only be visible within some approximately $180^\circ$ arc of longitude. There would be a side to the planet that never saw the moon. A closer Moon would produce stronger tidal effects, unless you also reduced it's radius by 1/9th. Alternatively, you could lengthen the Earth day to match the lunar month and keep everything else constant. On Earth, the only geostationary satellites that are stable are the ones at longitudes 75-West and 105-West, and a Moon would eventually settle down there over geologic time. These longitudes are particular to Earth's gravitational field, and would be different for another planet.
2. The Sun Synchronous Moon
A sun synchronous orbit maintains the same angle to the sun throughout the year. In other words, an orbiting body will appear in the sky in the same location at the same solar time each day. Everybody on the planet sees the orbiting body, but at a different time depending on their longitude and latitude. On Earth, sun synchronous satellites have short periods around 1-3 hours. For example, if a person on the East coast of the United States sees a sun synchronous satellite appear on the horizon at dawn, then every day it will appear at dawn (the same time relative to the Sun). Furthermore, since there is about a 3 hour time difference between the East coast and the West coast of the United States, a person on the West coast will also see the satellite at dawn when it comes around it's second orbit.
It is possible to extend that 3-hour orbital period a little bit if you fiddle with the mass and oblateness of the Earth, but it can't be extended to a single Earth day (I tried) without some nonsensical values.
Sun synchronous orbits need to be high inclination (near polar) orbits. However just like the geosynchronous orbit, half the locations on Earth will only see the moon during the day, and the other half only see the moon during the night. There is however a pleasant compromise: an orbit that is aligned perpendicular to the sun will follow the day-night terminator and will be visible to everyone on the planet only at dawn and dusk.
In a sun synchronous orbit, the moon would always appear in the same phase to the occupants of the planet, since it always appears at the same relative angle to the sun.
The L2 location gets you what you want, but the orbit is not stable to perturbations. If your planet is in a system with other outer planets, like Earth, then this will occur, and my intuition tells me it would take less than 100 planet revolutions if the solar system is like Earths.
There are some stable orbits that offer compromises, but someone on the planet will see the moon during the day, at least some of the time.