How many types of Pound Sterling are there and what is the relationship between them?

There is one Pound Sterling, which is represented by Bank of England, Scottish and Northern Irish notes and by Royal Mint coins (there aren't separate coins in England, Scotland and Northern Ireland).

Gibraltar, the Falkland Islands and Saint Helena (overseas territories) and Jersey, Guernsey and the Isle of Man (crown dependencies) each have their own pounds which are pegged at one-to-one with the pound sterling. They each issue their own notes (to their own designs) and coins (which have the same weights and compositions as Royal Mint coins but have different designs). In theory, they could be floated, in which case there would be an actual exchange rate; but the governments of those overseas territories and crown dependencies choose to maintain the peg at par.

Historically, the Republic of Ireland issued its own pound (or punt in Irish) and was maintained at par with sterling until 1979, when it was floated and there was an exchange rate until the Irish pound became part of the euro in 1999.

There is then the separate question of the acceptability of notes and coins. Legal tender is an irrelevance in a shop - it's always up to the shopkeeper what notes they will accept. It's only meaningful for settlement of a debt (e.g. in a restaurant or a taxi where the service has already been provided before payment) and even then, any note can be refused by any business other than a bank because they think it might be a forgery (banks will instead take and destroy the forgery). For that reason, many shops and other businesses will not accept notes that they are unfamiliar with, as they are concerned about forgeries and unable to tell genuine notes from forged ones. This doesn't just affect non-English notes in England and Wales (especially in smaller businesses and outside major cities), but also the Bank of England £50 note, which is not in wide circulation (few ATMs will issue £50s) and is often refused.

Scottish and Northern Irish notes are generally more likely to be accepted in England by larger businesses, in major cities and (for Scottish notes) close to the border - that is, in places where they are more likely to see them on a regular basis. For the more obscure notes, they are less likely to be accepted in businesses; you may have to take them to a bank to exchange for Bank of England notes. All banks are obliged to do this with Scottish and Northern Irish notes; they usually will in practice with crown dependency or overseas territory notes, but technically this is a foreign currency exchange for which they could charge. In practice, they don't.

Northern Irish notes are accepted more than they used to be; confusion with the Irish pound meant that many were concerned about accepting them and mistakenly taking a foreign currency that was worth slightly less than the pound sterling (like US and Canadian dollars). With Ireland switching to the Euro, this concern has faded.

CD and OT Coins, because they have the same weight and composition as Royal Mint coins, all work in vending machines. Cashiers in shops and other businesses often fail to notice that they aren't Royal Mint coins and will accept them as a result; some of them will accept them even when they know what they are, but most cashiers have been told not to, usually because there are lots of forged £1 coins in the UK and they are not trained to distinguish a non-Royal Mint coin from a forgery.

The first sentence of the gibraltarinfo.gi page you quote is oversimplified to the point of being incorrect. The currency of Gibraltar is the Gibraltar Pound, which is a separate currency but fixed at a 1:1 exchange rate with Sterling. People and businesses in Gibraltar choose to use both Gibraltar Pounds and Sterling for their own convenience. However, as the quote goes on to explain, people and businesses in the UK do not, in general, accept Gibraltar Pounds.

To some extent, there's a hierarchy: there are various issuers of Pounds Sterling, which are accepted in different places to different degrees; there are also issuers of Pounds that are not Sterling, and those are usually only accepted in the territory in which they're issued.

The acceptance of actual Pounds Sterling depends really on how familiar people are with the notes in question. I've never had any trouble spending Scottish notes in England or English notes in Scotland. English notes are very common in Scotland and, although Scottish notes aren't common in England, most staff in most shops see enough bank notes that they see a few Scottish notes and know what they look like. As I recall, the Clydesdale Bank prints fewer notes than the other two issuers of Scottish notes, so those are less familiar and might be harder to spend. The Scottish banks also issue £100 notes, which are probably impossible to spend in England, since there's "no such thing as a £100 note" (the Bank of England doesn't issue any note above £50) and, in any case, many shops refuse even Bank of England £50 notes because they're rare and an attractive target for forgers.

On the other hand, the only time I had a Northern Irish note in England, the first two shops I tried refused it and I took it to a bank to exchange it for a Bank of England note. Part of the problem here was that, at the time, the Republic of Ireland used a currency also called the Pound (or Punt; they now use the Euro) and my Northern Irish note was issued by the Bank of Ireland. "Look! It says '5 Pounds Sterling'" wasn't working for me.

When you leave Gibraltar, you'll probably have a mix of Gibraltar Pounds and Sterling. Spend the Gibraltar Pounds at the airport first because you won't be able to use those anywhere else.