Factsheet
The definitions of "transpiler" and "compiler" are blurry. Both of them do translate a program from one language to another language while keeping the behaviour.
We usually name it a "compiler" when it produces an executable binary. However, binary is just another language, which can be interpreted by a CPU. Every program is "executable" on the respective machine.
We usually name it a "compiler" when it produces a lower-level output than the input was, e.g. C to assembler. Or Java to Java bytecode. Or ES8 to ES5. Wait… is that really a different level?
We usually name it a "transpiler" when its output is on a similar level as the input, e.g. Python to JavaScript or the other way round. However, there always will be parts that use an abstraction available in one language that need to be "compiled" to a lower-level implementation in the other language.
So to answer your questions:
I believe a compiler is something very different from a transpiler.
No.
Is Babel a transpiler or a compiler, or perhaps offers both options?
Yes.
Or is the website's title 'Babel - The compiler for writing next generation JavaScript' simply incorrect?
No. That title focuses on next-generation features, i.e. higher-level abstractions that really need to be compiled into a very different output. Even though the output is still ordinary, mostly human-readable, JavaScript.
I am writing documentation relating to Babel, and I want to ensure my description is accurate
In that case, I'd use their own terminology from the official website. If you want to compare the tool to others, choose your own description.
Babel is a transpiler, which is a special type of compiler, so both terms are techincally correct. You may use either at your preference.
It's irrefutable that Babel is a source-to-source compiler (aka transpiler) since its source and target languages are both some flavor of JavaScript:
A source-to-source compiler, transcompiler or transpiler is a type of compiler that takes the source code of a program written in one programming language as its input and produces the equivalent source code in another programming language.
However, not everyone agrees that the distinction between the terms is helpful, so some people prefer simply, "compiler".
I personally like the distinction because, to me, it implies something about the difference in level of abstraction from machine langauge between the source (input) and target (output) languages. That is, typical "compilers" translate from higher-to-lower level languages, "decompilers" translate from lower-to-higher level languages, and "transpilers" translate between languages at similar levels of abstraction.