Larger teams can afford the level of specialization described here, but a smaller company (or a small internal team) may have a technical front-end developer who produces prototypes and live code, plus a non-technical designer who does everything else on the design side. If it's a really tiny team, there might be a single person who rapidly produces mockups and scripts for usability tests, then turns around a few days later and starts writing HTML/CSS/JS based on their findings.
I wish there were a better word than "unicorn" for people with a wider band of responsibilities. "Full-stack UX" sounds a bit silly too.
However, saying that someone who specializes in JavaScript is essentially the same as a back-end developer is inaccurate. The types of problems an experienced JavaScript developer deals with on a regular basis are fundamentally different than those that a back-end only developer deals with (not trying to imply that there isn't overlap).
I consider myself "Front-end developer". Actually I've been working and marketing what I"m doing with those 2-3 words for the last 10 years and I had been hired on a couple of jobs using this as a s a job title.
I don't think I'm a designer, though.
Designer is responsible for all UX tasks that don't involve writing lots of code, Front-Ender writes all client-side code "above the API." API code, devops, and DB admin falls to the server-side engineer. You might create a separate mobile role too, if you're working on a mobile client. If one of these roles gets contracted out, it's usually the designer or the mobile dev.
Then again, I'm most familiar with small teams. You see more specialization at larger firms.
As a front-end developer :
I've been doing JSPs, but I've rarely modified .java files.
I've been doing JS, but I would not be confident in pushing commits to the server-side code of a NodeJS project.
I've been doing REST API calls, but it would be hard for me to change database queries.
etc.
That's what I think I'm best at. ( You know as every developer it's not a big deal to dig into any problem, but professionals usually rely on experience ).