When I worked in investment banking, all our debt models were pegged to LIBOR. Rest In Peace! God bless the IB analysts scrambling to plug their models with replacement rates and weird hardcodes!
LIBOR + bank's minimum interest rate + a rate based on creditworthiness = your offered interest rate
Planet Money recently had a really good episode on how some banks are deciding on a replacement rate: https://www.npr.org/2021/10/08/1044598674/libor-pains
If anyone else was more accurate at this measurement, then they had arbitrage against those using the measure, giving those doing the initial measurement incentive to get it as right as is humanly possible, since they usually worked at places that use LIBOR to price things.
Since LIBOR underlied hundreds of trillions in assets, there are ample papers on all aspects of LIBOR, including those trying to see how well it was computed versus post outcomes.
It holds up well. https://scholar.google.com/scholar?hl=en&as_sdt=0%2C14&q=LIB...
The market - interbank rates - is quite small in terms of participants. I think the problem is the number was incredibly useful even if it wasn't accurate.
After the LIBOR scandal, the EU brought the benchmarks regulation (BMR) which says that interest rate indexes have to be based on actual transactions, just as you say. Euribor, the equivalent of LIBOR for lending in euros, was reformed to be based on transactions:
https://www.emmi-benchmarks.eu/benchmarks/euribor/reforms/
The administrator of LIBOR proposed doing the same:
https://www.clarusft.com/rfrs-libor-is-changing/
But in the end, US and UK regulators decided just to abolish it, in favour of overnight indexes based on real transactions (SONIA for pounds, which already existed, and SOFR for dollars, which was created for this purpose).
I believe this divergence happened because of differences in the lending markets. In the euro area, there is still a lot of unsecured term lending, which is what Euribor measures. But in the UK and US, this kind of lending has largely dried up, but there is a lot of overnight lending, so they chose rates which measure that. I don't know why the euro area is different to the US and UK here. It's possible that the euro market will evolve to be more like the US and UK, in which case Euribor will stop being credible, and the euro will also move over to its overnight rate, ESTR.
Another fun quirk is that SONIA and ESTR measure unsecured overnight lending, whereas SOFR measures "repo", which is essentially lending secured with government bonds as collateral. There is a sterling overnight repo rate, RONIA, but i don't think it's used much. I think repo volumes are higher than unsecured lending volumes; if that difference gets stark enough, perhaps sterling and euro regulators will force another switch, to the repo indexes.
LIBID is the rate at which you'd be willing to borrow. So the difference between the two is like bid vs ask.