There is a difference between a hire purchase agreement and a credit agreement and the definition given in the original post does not accurately describe a hire purchase agreement, often called HP.
To understand what an HP agreement is we need to know what a hire agreement, a credit agreement and a hire purchase agreement are.
A simple hire or rental agreement gives the hirer the right to use an asset so long as they make the hire payments but the owner retains ownership of the asset. The agreement can either be open-ended or subject to a lease with a time limit.
A credit agreement is one where money is advanced to someone so that they can buy an asset. There is a schedule of payments which, if it is complied with, means that the lender will be paid the amount leant plus a previously agreed amount of interest. The asset immediately becomes the property of the borrower but it is used as security for the loan and so can be reposessed if the borrower fails to make the payments.
A hire purchase agreement is a hire agreement for a fixed period of time with a right extended to the hirer to purchase the item outright for a fairly nominal sum at the end of the term. The size and number of the hire payments and the size of the final payment are calculated so that the hirer receives the price of the item plus a profit usually very similar to the interest that would have been charged had the agreement been a credit agreement. The point is that the asset remains the property either of the vendor or the HP company until the final payment has been made when the title to the asset passes to the hirer.
The etymology of the term "hire purchase" is, therefore, quite simple. A hire purchase agreement is one where the customer hires the asset for an agreed period then buys it for a pre-arranged nominal sum at the end of rental period.
Interestingly hire purchase agreements can be regarded as not violating strict religious objections to lending or borrowing money at interest (also known as usuary) and so are very similar to "Islamic mortgages" offered by Islamic banks.