Treat these as two separate claims. First you want to resolve the possible suit against you since that will hurt your credit--settle up with the other insurance, make sure you get a complete release of all claims signed by the other party before you pay anything. Once that is done with, then turn around and file a claim with your own insurance -- you may have to demand arbitration per your policy-- make sure you read your policy and follow all the steps described in it for the filing of an arbitration demand-if you miss anything they will simply refuse to participate.
Disclaimer: I am a lawyer licensed in the State of Illinois only, and I am not your lawyer (unless you have been in my office and signed a contract). This communication is not intended as legal advice, and no attorney client relationship results. Please consult your own attorney for legal advice. This is for informational purposes only.
If the damages are minor then you should try to resolve the lawsuit by the other insurance by settlement. You can then sue your insurance co. for reimbursement in small claims court where technicalities don't count as much as in regular courts. You bring your own phone bill to show that you called before renting the truck. That will support your allegation of incorrect information. It looks like at least DMV hasn't suspended your license for being uninsured but if the other insurance co. gets a judgment against you it will inform DMV and your license will be taken away (sometimes for a year even if you pay the judgment). So, settle if you can before the situation gets worse (assuming you can settle for a small amount).
Sign up to receive a 3-part series of useful information and advice about personal injury law.