Thanks for asking this question - as a solver of several, in my opinion the connect-wall is one of the most interesting and all-consuming of all puzzles. Although you comment Please don't say "Shouldn't this be meta?", there is unavoidably a PSE-specific component to the answer. However, I shall attempt to tackle your questions from a general standpoint too.
Q1a: What exactly is a connect-wall puzzle?
As the question tag here on PSE states, this is:
A puzzle involving a set of words which must be divided into groups such that the words in each group share some connection.
The first puzzle on PSE to have this tag was posted in May 2015, but this type of puzzle is not unique to the site - in fact, it is most well-known as a format used on the (fantastic) BBC quiz show Only Connect (2008-present) where it is known as a 'Connecting Wall'.
In the format established by the show, contestants are presented with 16 seemingly unconnected words and required to arrange them into 4 sets of 4 words, where each word in a set shares a connection. For example, the words 'sun', 'box', 'type' and 'jet' might form a group where the connection is 'can be followed by set' (sunset, box set, typeset, jet-set). Importantly, there is only one correct solution to partitioning the 16 words - although there may be several 'red herrings' which might fit in more than one category, there is a unique way to form 4 complete sets. (e.g. There might also be a group made up of 'cadger', 'hole', 'house' and 'threw', linked by 'initial letter can be changed to form an animal native to the UK' - 'box' from the other set could be 'fox', but must belong to the other group so that all starting words end up in a category of 4...)
Note that while the most common use of this puzzle on PSE is to find a connecting word which can precede or follow each of the words in the set, in the original show there were many more valid ways to group items - for example:
- Words which anagram to each other: STOP, POTS, POST and OPTS.
- Words which are anagrams of items from a connected list: RAISE, OLE, SPICES, BLAIR (signs of the Zodiac).
- Words which are homophones of items from a connected list: BEE, EYE, JAY, OH (letters).
- Words which contain words from a connected list: ROMAN, SCUBA, PIRANHA, MACHINATION (countries).
- and many others...
Also worthy of note is that here on PSE these puzzles nearly always require one single overarching connecting word to be found, which links the themes of the four main sets - this is not a requirement of the original TV show, but an interesting example of a convention that has quietly become the norm on this site!
Q1b: How do I write one?
If you intend to write a connect-wall puzzle in the PSE mould, work backwards:
- Decide on the word which will be your overarching connection.
- Come up with four words which can be connected by it. Remember that these words will need to be the connecting words for other groups of words in turn, so they cannot be too obscure.
- Once you have chosen four connecting words, devise four words which can connect with those. If you need a reference for phrases which can be linked by a particular word I recommend RhymeZone and the Word Finder facility of thefreedictionary.com.
- Most importantly, ensure that the resulting list of words has a unique solution. It's fine to include red herrings, but as soon as you have two words which can swap categories, satisfying both connections, (or indeed a chain of 3 or 4 exchanges) your connecting wall no longer has a unique solution, and the puzzle would be considered poorly thought out. (Note that, of course, some puzzlers can justify more obscure unintended connections between words, making unshakeable arguments for the partition they identified - if this happens, it's not the end of the world; more a sign that even the most carefully thought-out puzzles can be unpicked by a great mind on this site!)
One final tip on this: Always test your puzzle before publishing. Try to see if any swaps (as outlined above) exist - if so, change the words you thought of. Run your puzzle past somebody else first - if they solve it very quickly, change it (it's too easy); if they end up giving up, show them the intended solution - if they then argue that no rational human being would ever get this, change it (it's too hard!). Hitting that middle-ground of challenging-enough-but-gettable is your goal...
Q2: What exactly is a word-web puzzle, and how do I write one?
Q3: What are the differences/similarities between connect-wall and word-web?
I will answer these two questions at the same time... Seeing your thoughts about the differences between this type of puzzle and those tagged connect-wall, my advice is simple - stop thinking of these as two distinct puzzle types! :) Here's why...
If you search on PSE for the phrase "word web", aside from this question (and one which purely quotes another puzzle), there are only 4 questions that come up. Contrast this with the (at present) 64 puzzles tagged as connect-wall. If you delve further into this phrase you will find that 3 of the 4 instances were all created by the same user (hi @Mithical!), the earliest being published after 15 connect-wall puzzles already existed on the site. Meanwhile, the fourth of these puzzles - despite being called a "word web" by its OP - is actually tagged connect-wall!
The mechanisms of all 4 of these "word web" puzzles are very similar to connecting walls, although they may have a little additional flourish involved at some point in the puzzle solve - but this is a trait shared by many of the connect-wall puzzles also; many of these also deviate from the standard pattern in some element of their design. At the core of their design, however, there is very little difference between the essence of each.
To summarise, a "word web" puzzle is ultimately just a variant of a connect-wall puzzle, specifically a preference of one particular user of this site. At the heart of it all, similar rules govern both.
If after this you decide to create a connect-wall puzzle of your own: Good luck! (And I am always happy to test one out if required...)