Depending on the phone model, there is a setting that allows updating the CallerID.

In the extension settings, on PBXWare, look for the option “Send Remote-Party-ID” and “Use Remote Party ID”.

Also search for the option in the extension "Trust Remote-Party-ID" and toggle this option to Yes.

For a Yealink, go to its phone web administration interface (e.g.: 192.168.1.104), then in Account > Advanced > CallerID Source you must configure the option RPID-PAI-FROM.

Another solution is to inject the following provisioning parameter into the phone:

account.1.cid_source = 3

Under PBXware phones can be auto-provisioned by adding custom parameters in the UAD of the Yealinks in the field General user agent auto-provisioning template

For HTEK go to the phone administration interface, then in Profile > Advanced >  Caller Display Source > Configure the field RPID-PAI-FROM.