1.
A simple example: I could look at a sequence of numbers and make a new sequence of numbers in which each number is the difference between two consecutive numbers from the first sequence:
Then I could make sequence 3, with de differences between the numbers from sequence 2:
0, 2 (sequence 3: sequential subtraction, with as its object sequence 2)
Sequence 2 does with sequence 1 exactly the same thing (sequential subtraction) as what sequence 3 does with sequence 2. Both the operations carried out by 2 and 3 can be defined by the same name, sequential subtraction. We could say that 3 operates on a meta-level in relation to 2. We could say that 3 is meta-2. In logical language, we could define 2 as SeqSub(sequence 1), and 3 as SeqSub(SeqSub(sequence 1)). As you see, 2 is nested within 3.
On the other hand, consider grammar and normal language. Normal language describes the world, ideas, anything. Grammar describes, too, but it only describes language, not the world or anything. Grammar could be called meta-language. In this manner, we could call anything meta-x as long as x is a theory and meta-x is a theory about x, even if x and meta-x operate in different ways. X should be something vaguely similar to a theory, some abstract operation.
(You could even use meta- with things that aren't theories or abstract operations, but that is normally not done, except as a joke — suppose you had a brush to clean the floor, and a rag to clean the brush; then you might jokingly call your rag a meta-cleaner.)
Now what is the difference between grammar and sequence 3? We could say that grammar does not do exactly the same thing with language as what language does with its object, because grammar cannot, for example, refer to a physical thing directly. I think this what your quote means, the difference between identical operation on the one hand and similar-but-not-identical operation on the other. Sequential subtraction = sequential subtraction; grammar is a language, but language is not always grammar. In logical language, we could describe language as Describes(world), and grammar as GrammaticallyAnalyses(Describes(world)). They are nested, but in two slightly different ways.
Of course this distinction between "identical" and "similar" depends on definitions, which may be somewhat arbitrary. So I do not have full confidence in its strength and meaningfulness.
2.
I think I can feel what you mean, but I am not sure I'd phrase it like that.
I'd put metalogic in the second category mentioned above — similar to but not exactly the same as — but I am not sure. The reason why it is called "meta-" is that logic studies language and thinking, which makes logic an abstract operation and a theory; and metalogic studies logic, so that it is on a meta-level in relation to logic. Note that meta-x is always relative to its object: metalogic is not "meta-" in relation to, say, pottery.
3.
[Edited] In "metaphysics", the prefix "meta-" is used in its original sense in ancient Greek, which is here "after". Aristotle wrote the Physica, which were about the workings of nature: physis/phusis is Greek for nature. And he wrote the Metaphysica, which he called "The [books/bookrolls] about prime philosophy" — physics was the secondary philosophy. Later Greek scholars catalogued this work as "ta meta ta PHusika": "the things after the Physica", because they came after his Physica in their catalogue. Because his Metaphysica were about causality and other principles at work behind the physical world, it seems people later interpreted the "meta-" in Metaphysica as meaning "on a higher level than", and that is where our use of "meta-" came from.
4.
Stackoverflow is a website with questions and answers about programming. Meta-Stackoverflow is a website with questions and answers about Stackoverflow. So M-SO operates on SO the same way as SO operates on programming. In logical language, SO is SO(programming), and M-SO is SO(SO(programming)). You see how one is again nested within the other? That is why it is called Meta-Stackoverflow and not Newname.
They are two different things.
The dictionary is a book (or series of books) that lists all words in alphabetical order with pronunciation, definitions, classifications (noun, verb, adjective) and if it's bilingual (E.G. English-[other language]) it gives the translation.
The vocabulary is not an object, it's that collection of words used in a given language. It can also indicate the amount of words known by a certain person (E.G. John has such a wide vocabulary!).
EDIT: I wrote that lexicon is a synonym of vocabulary, and it is, but Rhodri made me notice that it's also a synonym of dictionary. So I investigated a bit and, as I was suspecting, it has a restricted use, which is still right though... From the OED:
A word-book or dictionary; chiefly applied to a dictionary of Greek, Hebrew, Syriac, or Arabic. The restricted use is due to the fact that until recently dictionaries of these particular languages were usually in Latin, and in mod.L. lexicon, not dictionarius, has been the word generally used.
Best Answer
Asking for “formal definitions” is fraught with peril: who is the recognized authority? In any event, “searching around the web” isn’t going to do you much good. If you want to know what a dictionary says about a dictionary, then you shall have to actually use one. :)
All these words have multiple senses, depending on the context and domain:
Dictionary is not so simple as you seem to present it; your discussion of “mapping” suggests you are thinking more of a polyglot-type dictionary or else a computer dictionary, but neither of those are the most common current use for the word, which has a long history with many senses and subsenses.
Lexicon was historically used for “look-up”–type dictionaries for languages like Greek and Hebrew, but also has a particular sense in linguistics.
Vocabulary has at least four main senses, of which you allude to at most one. Sometimes it is just a word-list, but sometimes it is more than that.
Based on the assumption that the curated Oxford English Dictionary is a better primary source for words’ meanings in English than the crowd-sourced Wikipedia entries you stumbled upon, here’s an excerpt of what the OED says about all this.
I suppose you can take these for “formal definitions” if it pleases you to do so, but really they just document matters as the editors of the OED saw fit to do.
It should come as no surprise that the OED takes especial care delineating the historical and extended uses of its own name, dictionary. I’ve omitted a good bit of historical discussion from its excerpted notes below; see the real dictionary if you need that.
Truth be told, the note at the end of dictionary sense 1a is probably all you need, but I’ll include a little more than that just in case. The important paragraph that distinguishes one from the other I’ve highlighted in bold, but the other senses show how domain-specific use of these terms can be, well, more specific in that particular domain.
Oh, and I threw glossary and word-book in because I knew you’d just come back and ask about them, since they are referenced in the other entries. As you can see, there is a lot of overlap in these words, particularly those defined in terms of one another. So even the OED contains what you might call circular references; this always happens with close word-sets like these.
However, there are also fine distinctions to be found and made, should one wish to do so.