AT&T upgrade jargon, if anyone knows (for iPhone 6)

Question

I have an iPhone 5C, which has served me pretty faithfully for the time that I've had it. Sadly, over the weekend it was dropped onto a hard floor and broke. (It had been in a case, which I had removed for cleaning...of course.) The screen is intact but full of flashing lines instead of being visible. Also the light on the back won't turn off. I've got a picture of what it's doing now if anyone is interested. I'd like to get an iPhone 6, if possible, since iOS 8 seems to be rather creaky on the iPhone 5C.
My phone is on my parents' plan, so my mother told me to check if an upgrade is available. I went to AT&T's site, and this is what it told me:
"Congratulations! This line is eligible for an AT&T NEXT installment plan upgrade. Or upgrade with a 2-year service agreement and a $40 upgrade."
Forgive me, but I have no idea what this means. I mean, I do, but I don't trust AT&T any farther than I can throw them, and my inherently suspicious  nature makes me suspect that they're not being entirely honest, i.e., they're trying to bait me in by making me think an upgrade would be free or just a one-time $40 payment. What follows is a list of several phones listing how much they would cost a month.
What's the difference between an "installment plan upgrade" and a "2-year service agreement" costing $40? I figure that what AT&T is trying to sell me is a new phone with a two-year contract on top of the one my family already has.
Again, forgive my ignorance if I am way off, but I'd just like to understand what my options are before I march into the AT&T store to get a new phone. Of course, if anyone thinks that it might be cheaper to take the phone to the Apple Store to have it repaired, tell me. The internal workings seem to be alright, as I plugged it into my MacBook Pro to back it up. This is the state it's in now:

Answer 1

iPhone 5C out of warranty replacement is $269.
Or upgrade with a 2-year service agreement and a $40 upgrade."
This simply puts you into a 2 year extension of you current AT&T contract.
You get to pay $199 for a new iPhone in addition to the $40 fee from AT&T is the “upgrade/activation” fee (it’s simply extra $$ they can get from consumer).
Just tell them you do not want to pay it or want it refunded. They can easily do this.
This line is eligible for an AT&T NEXT installment plan upgrade.
The “installment plan upgrade" (aka the NEXT plan) is you simply purchase a new phone at full price and the payments are spread out over 20 or 24 months.
You are not stuck on a 2 year contract. In addition, your service is usually $25 less per month (but you pay an extra +$25 for the phone).
Either way, you are paying the same full price for the phone (whether it is directly for the phone or extra $$ on the 2 year contract).