You are correct in your interpretation of "bill of collection" in this case - it is a type of invoice. There is a lot of technical legal language involved in explaining the difference between a bill of collection and an invoice, but in simple terms a bill of collection is sent directly to the bank where the person/entity has his/its account, and not directly to the person who accrued the charges. Some businesses use this manner of debt collection because it has a separate set of rules under the Uniform Rules for Collection.
"Trial-minded" refers to a thought process that anticipates all the legal aspects of a situation. It is presumed that the situation/case will eventually go to trial in court before a judge and jury, so care will be taken to collect evidence and present the facts with that in mind.
Student or Learner