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.
Best Answer
Most words have multiple meanings depending on context; professional is no different.
Profession, according to Etymonline.com, originated in one's declaration to join a Catholic religious institute:
For the adjective form professional,
Professor, referring to a university instructor of a certain rank, is related, but is borrowed via Old French directly from Latin:
So it seems the sense of professional as one in an occupation requiring special training, skills, or education is the original one, and it is only loosely related to professor.
And as with most words, context is important in how you would use professional.
In sports parlance, for example, a professional is one who is paid to play a sport as opposed to an amateur who trains as a pastime. This notion is independent of any skill level or training, as evidenced by the controversy over the treatment of student-athletes in the NCAA, and by the New York Mets. Between the two is the world of semi-professional sports, where competitors are paid, but do not train and play as a full-time profession, and must take at least part-time employment to meet their expenses.
In the military, in contrast, a professional army is comprised of enlistees who make the military their full-time career, as opposed to private citizens who volunteer or are conscripted (a citizen army).
In economics, political science, and other social sciences, professional refers to a class of workers with advanced education and specialized training: medicine, law, engineering, academia, and so on. By this definition, an administrative assistant would not be a professional even though he is paid to do the job, and may wear a tie and work in an office; the job requires no specialized degree or advanced training.
In still broader contexts, professional may be synonymous with white-collar worker, including not just doctors and lawyers but people like consultants, analysts, or really anyone who works at a desk. When economic development boards and fashionable apartment buildings promote their friendliness to single "young professionals" it is in this sense of the word.