So I have recently started a Biochemistry course online. I am learning all about proteins and cell structures and purification techniques.

I am now thinking that the SR tradition of having "biotech" cover trauma first aid seems kind of erroneous.

Shouldn't it be "Trauma First Aid" or "Combat Lifesaving" or something that lets you stabilize a teammate, and then a separate "Biotech" skill if you want to go and grow a replacement organ later?

Biotech is the Skill Group which covers the Skills of Biotechnology, Cybertechnology, First Aid, and Medicine

The first, second and third editions of Shadowrun had a Biotech skill that covered everything from first aid to medicine and cybertechnology. I'm not sure if the authors avoided simply calling to "Medicine" to prevent PC from claiming full physician capacity after spending a few points, if they wanted it to sound cyberpunk and all that, where even the most basic medkits were to contain all sort of advanced healing technology or if it was just to lump cybertechnology in (causing endless arguments on whether Biotech, Electronics, B/R Electronics or B/R Computers should be used to physically access headware memory...). Also, it seemed less counter-intuitive to make cybertechnology a subset of biotechnology at a time bioware had yet to be introduced as an alternative to cyberware.

The fourth edition divided the skill into three different skills, Cybertechnology, First Aid and Medicine (there is no Biotechnology active skill), and retained the name for the Skill Group that features them.

In a nutshell, yes. But then, the game was really focused on Shadowrunners, and what Shadowrunners do. Shadowrunners don't generally sit in a physican's office and build cybertech, so there wasn't much focus on there. But come M&M, they should have gotten their act together and explain maybe why a doctor goes to school for 10 years so they can get an 8 in a single skill.