How are the axes of the laser defined, i.e. what is vertical?

The vertical direction is the so called growth direction.

In practice, you have a device in front of you, it is mounted on a copper carrier. The carrier has one or two ceramic pads carrying the bonding wires. The pads are yellow on top due to a layer of gold, and white around it and on the sides (colour of the ceramic). If these pads are placed upwards, the vertical for the laser is the same as the observer vertical direction.

If there are two ceramic pads present, they are named as follows: Looking onto the front facet with the laser placed as described above, the pad left of the laser chip is called "down", the one right of it "up". If no configuration is specified, the "down" pad is used.