Cut the dead part off and keep the rest as it is, where it is.

I vote for neither choice. Most wild avocados grow upwards of 40 feet. Commercial trees are pruned but still get half that size. Avocados as trees need tropical climates. Furthermore, not all varieties are self-fruitful, so you might never get fruit even if you have the room for a tree.

I recommend you grow your avocado as a houseplant. That means cutting it back to produce multiple stems. The best time to prune is late winter or very early spring, but I think you should cut the dead stem back to above the best outward facing bud now. Be sure to sterilize your cutting tool before use by wiping down the blade with a 1 part bleach to 9 parts water solution. Cut back the other stem at the regular time. Never cut more than 1/3 the length of a stem. Prune the tops of the plant each year to keep it small and shape it.

Avocados prefer free-draining soil with a pH of 6-6.5. Water twice a week, making sure to soak the soil. Add a balanced liquid fertilizer to the water at least 3 times a year in early spring, summer and fall. Place the plant in a sunny window, and mist the leaves regularly as the plant prefers higher humidity.

trim the dead stem, keep the green on as is. If it doesn't branch out when it gets 4 foot or so, think about chopping its head off. Give it a bigger container if it is beginning to get root bound. I am growing one in a container (1.5 year-old, 2.5-3 foot) and am not giving up on getting it to fruit in 4-6 years. It is not out of the ordinary for 5-7 year old seed grown avocados to begin to bear fruit. Most are self fertile (perhaps with some intervention for a container grown one). We shall see...Meanwhile I have a houseplant.