Tomasina, your drawing does not match my description. For example, I never suggested the use of I2C, a standard not meant to cover long distances. 1-wire and I2C are very different animals.
The simplest solution (with the most wires) is to use a common GND wire and a individual wire for each light strip coming from the teensy. The teensy uses a PWM chopper and a Darlington transistor or FET to duty cycle the 12VDC power supply running to the light strip. That in turn allows you to turn on and turn off each light strip with minimal additional circuitry. Depending on your configuration, etc. you might be able to do this with flat cable, for example. All nodes come together on one end, then split off individually at the other end of the cable. Given the distances and number of wires needed, perhaps not the best solution.
The next solution is to use a serial bus in conjunction with one small MCU per light strip to do the PWM. You'd need 4 wires, 2 for power, the other two for the bus. This may or may not work due to signal attenuation, noise, and so on. However, it is likely the cheapest solution with a distributed bus architecture since every MCU out there has a serial port.
However, as Paul and I point out, you are likely going to need RS485 chips to create a true differential signal that is more noise immune, that can cover long distances, and so on. You'd still need 4 wires for your bus, but now you'd also need to add a RS485 chip for each lighting strip (in addition to the MCU there). If you all you do is plan to send signals from the master and never reply back from a slave, you could also consider receiver-only chips on each lighting strip like the MAX3280E.
As I mentioned earlier, there are established lighting protocols out there like DMX that use RS485. You may want to consider studying them and a DMX-capable solution could allow you to add other gear as well.