What's the difference between an inverted index and a plain old index?
One common use is "...to allow fast full-text searching."
The two types denote directionality. One takes you forward through the index, and the other takes you backward (the inverse) through the index. That's it. There's no mystery to uncover here. Otherwise the two types are identical, it's just a question of what information you have, and as a result what information you're trying to find.
To address your inquiry, I don't think there's actually a way to know why the use is what it is today. The only reason it's important to define which is forward and which one is inverted is so that we can all have a conversation about them, and everyone knows which direction we're talking about. Think about the terms "left" and "right": they are relative. Which is which doesn't matter, except that everyone needs to agree which one is "left" and which one is "right" in order for the words to have meaning. If, as a culture, we decided to flip left and right, then you'd have the same issue figuring out what a "right turn" vs a "left turn" is since the agreed upon meaning had changed. However, the naming is arbitrary, so which one is which (in and of itself) doesn't matter - what matters is that we all agree on the meaning.
In your comment where you ask, "please don't just define the terms", you're missing the point, and I think you're just getting hung up on the wording when there is absolutely no difference between them.
For the benefit of future readers, I will now provide several "forward" and "inverted" index examples:
Example 1: Web search
If you're thinking that the inverse of an index is something like the inverse of a function in mathematics, where the inverse is a special thing that has a different form, then you're mistaken: that's not the case here.
In a search engine you have a list of documents (pages on web sites), where you enter some keywords and get results back.
A forward index (or just index) is the list of documents, and which words appear in them. In the web search example, Google crawls the web, building the list of documents, figuring out which words appear in each page.
The inverted index is the list of words, and the documents in which they appear. In the web search example, you provide the list of words (your search query), and Google produces the documents (search result links).
They are both indexes - it's just a question of which direction you're going. Forward is from documents->to->words, inverted is from words->to->documents.
Example 2: DNS
Another example is a DNS lookup (which takes a host name, and returns an IP address) and a reverse lookup (which takes an IP address, and gives you the host name).
Example 3: A book
The index in the back of a book is actually an inverted index, as defined by the examples above - a list of words, and where to find them in the book. In a book, the table of contents is like a forward index: it's a list of documents (chapters) which the book contains, except instead of listing the words in those sections, the table of contents just gives a name/general description of what's contained in those documents (chapters).
Example 4: Your cell phone
The forward index in your cell phone is your list of contacts, and which phone numbers (cell, home, work) are associated with those contacts. The inverted index is what allows you to manually enter a phone number, and when you hit "dial" you see the person's name, rather than the number, because your phone has taken the phone number and found you the contact associated with it.
They called it inverted just because there is already a forward index. Take the example of search engine, it composed by two parts: the first part is "web crawler and parser" which build a index from document to word, the second part is search database which build a index from word to document. Because of the first index exist, we naturally call the second index as inverted index.
If you name the TOC (Table of Content) of a book as index, then you should call the index at the end of book as "inverted index". Or, in other side, you can call the TOC as inverted index.
normalocity has already wonderfully differentiated between a forward and an inverted index but for the question of why one is called a forward index and the other an inverted index, maybe this is why they are called that way---
Taking example of search engine crawling and indexing (or building index for a book), a forward index can be built simultaneously while you are crawling the web pages(or reading the book) or going forward. So if you have 10 webpages to crawl(or 10 chapters in a book) you can crawl the first webpage(read the first chapter) and then make a list of words which appear in the webpage(words which appear in the chapter) and continue this process for other webpages(other chapters) so by the time you have crawled all the 10 webpages(read all 10 chapters) your forward index is complete with each webpage(chapter) pointing to a list of words it contains.
But to make an inverted index you have to crawl all the 10 webpages(read the 10 chapters) and and then take each word from each documents list and figure out which documents contain that word. So this is like going backward once you have crawled the webpages(read chapters of the book). So its called an inverted index.
This is just my speculation.