I generally think of a "full-stack" developer not as someone who is proficient with all of these tools, but rather one or two in each area, and has the necessary background to easily adapt his experience into another toolkit.
For example, someone who is proficient in:
Could quickly learn all the other things in your original list as needed.
Could easily adapt to work with Vue. Babel + Eslint + Test Framework would likely be picked up on accident along the way.
Back-end is likely the one that would be trickiest, mainly due to the plethora of available languages and frameworks, but once you've worked with one or two toolsets, the core concepts still translate fairly well between heavy hitters.
Once you've got basic MySQL, or even SQLite down, with some focused studying you could probably adapt to whichever SQL-based database your employer is using.
Document and in-memory stores tend to be fairly easy to understand, comparable to any medium-sized library.
Web Application Architecture is, well, a pain. Every platform has its own tools all over the place that may or may not work well together. Occasionally concepts translate, sometimes they don't. But still nothing that good documentation and messing around can't help wrap your head around.
To reiterate, In my opinion a full-stack developer is someone who has considerable experience working at all points in the stack, and feels comfortable with one or two tools. They're valuable not because they know everything you'll be using already, but because they can adapt quickly based on the background knowledge they've accrued.
Based on that understanding, I feel fully confident in calling myself a full-stack developer. However, I prefer the term "Developer" or "Software Engineer" just because they're slightly less maligned, fairly non-specific, and allow me to apply to jobs based on how well I think I can meet requirements.
You have to be careful with the term "engineer" though. It is a title which is acquired by going to and finishing an education. This might not be true in some countries.
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