My question is if instruments have their own interest rates, or if there is one global interest rate affecting all.
Well yes and no. Different instruments and different loans will often (but not always) have different interest rates. If you walk with your business proposal into bank based on your credit history and other factors you might not get the same interest rate as your friend. You can just easily verify this by looking at yields of various bonds of US firms, you will not
However, all interest rates are somewhat connected. This is due to competition and supply and demand interactions. If there is perfect competition then you would not expect 2 loans with exactly same terms maturity and so on to have different rates of interest. In real life markets and might not be perfect and competitive, but even in imperfect markets that are not competitive firms cannot set their prices in arbitrary fashion. Even monopolist can charge arbitrary prices.
Due to this similar debt will face similar interest rate. If for example bank A offers 10y \$1000 loan at 3% and bank B offers \$1000 10y loan at 5% (ceteris paribus), then why would any customer ever purchase the loan from bank B?
However, most debt is highly heterogenous. Bonds of Apple have different risk than bonds issued by GameStop and consequently will have different yields as GameStop will have to pay higher interest to compensate its lenders for taking the extra risk of borrowing to them.
As a third scenario, I suspect that there is actually one "master" interest rate on which all others depend, based on this article: https://www.bankofengland.co.uk/knowledgebank/what-are-interest-rates.
Yes, there actually is a "master" interest rate this is actually not third scenario, this is parallel to the discussion above. Most economies do not practice free banking, but central banking systems.
In central banking system, central bank (usually government institution), will serve as an lender of reserves to commercial bank. As a consequence central banks can affect all interest rates across whole economy because the central bank rate at which private banks can borrow determines in turn the rate at which individuals can borrow from private banks.
Commercial retail private banks earn profit primarily on intermediation margin which is the difference between what they have to pay to central bank or other people who deposit money with and them and the rate at which they lend money to individuals who want to borrow (see wider discussion of this in Firaxis and Rochet Microeconomics of Banking Ch 1-3). If private banks can borrow more cheaply at central bank well then they can also lend money more cheaply to individuals, and competition will force them to adjust interest rate. All debt instruments compete with each other so when you can borrow money from private banks more cheaply other interest rates have to go down as well. So central bank's interest rate can be thought of as a "master" interest rate, in a sense that it affects all other interest rates, but not in a sense that all interest rates are equal to it.
When people generally mention "the interest rate", are they talking about a particular rate or a collection of rates? For instance, is the savings rate equal to the loan rate? (I always though that loan rates > saving rates).
They are talking about particular interest rate. Whenever you see in some textbooks supply and demand graph with one interest rate that assumes that all debt on that particular market is the same.
Is "the interest rate" in the context of bonds with comments such as "interest rates influence bond prices" the same quantity as "the interest rate" in the context of banking e.g. "loan borrowers will have to pay higher interest rates"?
Again ceteris paribus if we have a 10 year bond with the same risk same duration and everything else as 10 year loan, and we assume markets are competitive, then yield to maturity on that 10 year bond must be equal to the interest rate on that 10 year loan (with bonds we talk about yield to maturity since they are fixed income securities and have fixed coupon payments but changes in the bond price create an implicit interest rate we call yield to maturity). However, if we talk about debt that is of different risk and so on the interest will not be same.