I’ve tried to book directly through the hotel site. And when I’ve informed the reservation desk the rate is cheaper through an OTA they tell me to book using the OTA. That was a 20% commission they could have kept or given to me as a discount.

I guess I don’t know how agreements work. Seems like OTAs are more flexible with pricing compared to the hotel staff.

Because the rate is not released by the hotel. Some agents undercut the rates, or floates a rate that is supposed to be marked up, naked (without mark-up) to the web. Asking the guest to book from the OTA is to see what voucher the hotel receives, and catch that agent as the culprit who undercut the rate without hotel's consent. The agreement with hotel and agent is that hotel can refuse to honour the voucher. In this case, the agent has floated a rate not discussed with the hotel. Hotel asked the guest to go ahead and book if there's a lower rate in OTA. Guest books, hotel has the right to refuse to honour the voucher.

I think the difference is in mainly in booking conditions.

For instance a lot of OTA will push non cancellable rooms for lower prices. The hotel probably doesn’t want to deal with these customer issues directly, while the guaranteed benefit is attractive.

The OTA can buy & manage it's own stock of hotel rooms in advance for hotel X and then sell them on for $whatever, $whenever.

Not every OTA is just a proxy. Some maintain their own stock and do forecasting/yield management etc..

This. I book quite a few hotel stays throughout the year and you will definitely get perks and resort credits when booking directly. Just ask for it. They don't want you hanging up the phone and going back to Kayak which may lead you to another property while deciding.