We can classify stablecoins into several categories: centralized or decentralized, supported by or without a cash cover, or linked to gold/bitcoin/dollar/euro. All of these classifications can include many pairs of stablecoins.
You are right, there are different criteria for classify stablecoins.
One criterium is, what are they backed/collateralized by, which is the criterium used by user Brewmaster, which I commented on.
In fact, there is no gold, but rather they are currencies supported by a financial cover, whether it is the dollar or the euro, or an attempt to be an asset like gold, but in reality there is no guarantee other than promises.
Well in some cased It's more than just promises: in some cases there is the
certification of an auditing company, which means the auditing company put their good name in it, losing a worldwide reputation if they certify what is not true.
For example, this is the
audit of the gold stocks of Kinesis conducted by Bureau Veritas, an internationally renowned auditing company founded in 1828.
Kinesis explains also
How does an audit work:
"At a designated time and date, independent officers from Inspectorate International enter all of Kinesis’ global vaulting facilities and physically tally the quantity of gold and silver held.
Shortly thereafter, the inspectors checked the total against the quantity reported by Kinesis’ strategic partner, ABX, in the inventory report on a prior date. Therefore, the quantity of gold and silver held in our vaults is checked against the record of all KAU and KAG in circulation, stored on our blockchain.
As well as recording the quantity, the auditors assess the quality and weight of a random selection of gold and silver bars. For this purpose, the inspectors carry out tests to assess the weight and fineness of the gold and silver.
Additionally, the auditors assess whether or not the quality of the storage facility was satisfactory."
I think this is something more than just promises

There were several projects based on gold, but most of them were not famous, and I think that one of the longest of these projects that is still working today is https://gold.storage/
This project is interesting. I'll look into it.
Thanks HB