In certain cases where you have repaid some installments of a previous student loan independently, you would be given the authority of branching away the Co-borrower. This is known as Co-Borrower Release and by doing so, you ensure that the loan repayment is entirely yours. This also puts you in a position where you are not hit by the Credit History of your Co-Borrower. Both of you are separate entities speaking from the loan perspective and you would hence need to take charge of repaying your loan off.
There are many financial institutions that offer private student loans with bad credit. The only fuss in the deal is the requirements of the financial institutions. Most entities that I know of providing this service would require a Co-borrower or a Co-Signer. This means as a student you and your parents would need to sign on the loan agreement agreeing to repay the loan.
"Insurance" organizations, who collect premiums for providing either life or property/casualty coverage, created their own types of loan agreements. "Banks" and "Insurance" organizations loan agreements and documentation standards evolved from their individual cultures and were governed by policies that somehow addressed each organizations liabilities (In the case of "banks," the liquidity needs of their depositors; in the case of insurance organizations, the liquidity needs associated with their expected "claims" payments).
For commercial banks and large finance companies, "loan agreements" are usually not categorized although "loan portfolios" are often broadly characterized into "personal" and "commercial" loans while the "commercial" category is then subdivided into "industrial" and "commercial real estate" loans. "Industrial" loans are those that depend on the cashflow and creditworthiness of the company and the widgets or service that it sells. "Commercial real estate" loans are those that repay loans but that depend on the rental revenues paid by tenants who lease space, usually for extended times. More granular categorizations of loan portfolios exist but these are always variations around the larger themes.
sample loan agreement between two parties
loan agreement between individuals
loan agreement form