I think all of this has been said above, and sounds like you've already started some of this too. Step by step this is what has typically always worked for me:
1. Document everything as clearly and concisely as possible. Fill out that accident report.
2. Call your own insurance and get on the record with them regarding what happened, get them the video, etc. Make sure your story is consistent. This way they are not surprised if they get a call from the other party claiming its your fault.
3. Call the other party's insurance and initiate a claim with them. Provide them all of the same details and video evidence. Make sure your story is consistent. This way they are not surprised if they get a call from the other party claiming its your fault. You can technically make your own insurance company do this, but it's your truck and your time, it's just more important to you.
4. Get to the body shop ASAP, get documentation for what needs fixing and costs. Make sure it's consistent with your story as well. Send this information to both insurance companies.
5. Check in periodically with both insurance companies and remind them of their responsibility to resolve this ASAP.
When you step up and initiate everything it goes a long way in your favor from my experience. Insurance companies don't like having to explain for their customer who hasn't contacted them and refuses to return their phone calls.