The day after my car was written of by a third party, a rep from their insurer turned up at my house! They wanted to find out if I would be making a Personal Injury claim and to make me an offer on the doorstep. (I sent them away).
Credit hire is where a company like DriveAssist or some one provides you a hire car on a credit basis. You are not expected to pay upfront (hence credit agreement), cost is billed to the insurers, but that cost is much higher than usual market rates. Hence insurers want to get you out of hire asap.
They may also be calling to see if you are injured. Most of the major insurers have teams who will deal directly with simple injury claims to save you the cost of instructing a solicitor.
Because it is likely to be cheaper than your own insurers paying for it.
I was rear ended a few years back and the other driver’s insurers were on the phone admitting liability and offering to get my car fixed/pay for car hire before I’d even had a chance to report it to my own insurer.
We had the same, hit by a driver clearly at fault (on his mobile ). His insurance co were ringing to find out if we were going to make a personal injury claim and to make us a settlement offer. We ignored them because we (amazingly!) weren’t injured so didn’t want to claim.
I have recently had a bump with the other driver claiming I had done his bumper in. No damage to my car. I thought he had brushed my wing mirror.
Some company called me on behalf of him to ask my version of events. I refused to speak with them and prefer to deal with my insurance company as he has made an allegation against me. I have dash cam footage and he is apparently making a claim for whiplash.
He drove from the scene, parked 200 yards away and he then came to my car claiming I had damaged his bumper.
There were no injuries.
OP, no harm in speaking with the other driver as liability has been established.
In my case, liability has not been established so I am sticking with speaking with my insurance company.