Let's just talk about USD specifically. (Other currencies work pretty similar, just with different central banks.)

The "single source of truth" is the Federal Reserve. Member banks (about 10,000 US domestic banks) can hold reserves directly at the Fed. Any reserve bank can directly transfer Fed reserves to another member bank or to the US Treasury using FedWire. The Fed "creates money" simply by crediting a member bank's reserve.

This is done in two ways, one is by paying interest on reserves. So if you're a member bank holding one million in reserves, the Fed simply bumps your balance up every night to reflect your interest payments. The other is by the Fed conducting "open market operations". When it wants to stimulate the economy by "providing liquidity" the NY Federal Reserve trading desk will buy treasury bonds and pay for them simply by crediting the balance of whichever member bank sold the bonds.

Member banks themselves can "create money" in the form of demand deposits. If you have one million in your checking account at Chase bank, all you really have is a promise by Chase bank to give you or whoever you want to send money to one million dollars. Essentially money is created the same way the Fed creates money, your checking account is nothing more than a credit in Chase's database.

However Chase's ability to create money is highly regulated and constrained. Banks have to conform to strict capital and leverage constraints. The OCC and other banking regulators will require that Chase hold enough assets in high enough quality to cover its deposit "liabilities". Some fraction of those assets will be dollars held as reserves at the Fed. Some will be bonds or loans, i.e. promises by other high-quality actors to pay dollars. Some of those assets could even be risky stocks, low quality bonds, commodities, etc. However the "risk-weighted" leverage ratio quickly scales up with risk.

In exchange for this regulation, deposits held at Chase or other member banks are treated as equivalent to Federal reserve dollars. That's because if for some reason, Chase is unable to pay back its depositors when they demand their money back, the Federal Reserve will "create money" to meet the obligations. This is essentially why bank runs no longer exist in modern banking.

Then there's the concept of "money markets", which is short-duration, high-quality assets that are essentially treated as "money-like". Often this tends to be short-term commercial paper from high-quality corporate issuers. A promise from Apple to pay $1.00 in 30 days is treated as "almost a dollar" by most of the market. In this sense, Apple can also "create money". But it's highly constrained, because it would quickly lose its credit rating or breach covenants in its longer-term bonds if it did so.

Finally there's the concepts of "eurodollar deposits". These are dollar deposits held at non-American banks but denominated in dollars. For example you could hold $1 million dollars at Mitsubishi bank in Japan, and you have a claim against Mitsubishi to pay out dollars whenever you want. Mitsubishi is not an American bank, and therefore is not directly regulated or insured by the Federal reserve. In some sense this makes eurodollars slightly less safe than dollars held at domestic member banks.

That being said Mitsubishi is regulated by Japanese banking regulations and implicitly insured by the BOJ (central bank of Japan). Mitsubishi will also make sure its dollar liabilities are never that large relative to its yen-assets. In the case of Mitsubishi failing, its highly likely the BOJ would bail them out by creating yen in the same way the Fed creates dollars. Some of that yen could be converted to dollars on the open FX market, and make the dollar depositors whole. Hence eurodollars are essentially just as safe as regular dollars as long as they're inside another high-quality banking system.