...so someone can't come along later and say you didn't pay it
That's why you get it in writing. That is especially important when things have gone to (or are about to go to) a third party collection agency. Old debts get traded around all the time, and it's absolutely possible for your old settled debt to pop up with another collection agency. You want to have a letter to fax them, or to wave around in court. Send them copies, obviously.
It's also important if you are brokering a bargain other than "just paying them in full". Because you may need to hold their feet to the fire to keep their side of the deal. In my experience, if you settle for a fraction, it is very hard to get them to say on your credit report that you paid in full. They will report a weird code word that sounds sorta like "paid in full" but is actually a dogwhistle to industry insiders that you settled.
Settling for a fraction isn't as valuable as you think
Because they will issue you a 1099-C for the forgiven part of your debt, and you will have to pay taxes on that money you were "gifted".
There are exceptions, but you have to fight for them by filing a Form 982 to counter the 1099-C.
So you'll get bit for, say, 25% of the forgiven debt on the Federal side, and maybe 8% on the state side, varying wildly by your income and state. It still beats paying it in full, but not by as much as you'd hoped -- and then, you get slammed with this big lump-sum extra tax, because no withholding is done.
If the amount is large and your financial stars misalign, You might even need to prepay it in a quarterly estimated tax payment, or face penalties for under-withholding. You are expected to know when this applies.
Given all the tomfoolery and monkeyshines associated with IRS effects, I am cautious of negotiated settlements. Better (emotionaly, not financially) to pay it in full then tell the IRS "can't touch this!"