Proof of Living Arrangements While Staying in UK?

I am 17 years old, and I am traveling to Manchester, England from Boston, USA on the 4th of July (I know, what a Benedict Arnold, lol). I will be traveling alone, and I will be staying with a friend at her home in Nottingham, England during my visit. I have a Passport, a Flight, Parental Permission, and Sufficient Funds for my stay - all the things that I need.

However, I'm a bit confused on what exactly I need to show Customs in order to prove that I have Parental Permission. On the UK Border Agency website, it said:

"Unaccompanied child visitors
If you will be travelling to the UK without an adult (someone over the age of 18), we will treat you as an 'unaccompanied child visitor'. Your parent or guardian will need to provide:

their written consent for you to travel to the UK;
their full contact details; and
evidence that suitable living arrangements have been made for your stay in the UK."

I'm a bit hung up on this, and I have a few questions which I was hoping someone could give me advice on. I've asked friends who have traveled internationally in the past, I've done research online, and I am also going to call the British Embassy in Boston tomorrow. But, I was just wondering if I could get a second opinion on this, just to calm me down, haha.

My questions are:

1. By "written consent" do you think they actually mean "written?" Or would a typed out letter with an ink signature suffice? I know it sounds like a bit of a stupid question, but I don't want to get all the way there and then be told that my understanding wasn't correct.

2. What exactly would I need to show as "evidence that suitable living arrangements have been made?" Does that just mean I need a letter from my mother saying "Yes, I grant my daughter permission to travel from Boston to Manchester, etc. etc. etc. I understand that she will be staying with a friend, [insert name here], at the following address, [insert address here]. This individual can be contacted at [insert contact details here]." Or would I need further evidence from my friend's mother?

Considering how close I am to my date of departure, what must I show as evidence for my living arrangements other than a signed letter from my friend's mother (since obviously it's too late for the original letter to be mailed to me)? Could she send me an email or send me a scanned copy of her letter?

Thank you very much in advance!

1. By "written consent" do you think they actually mean "written?" Or would a typed out letter with an ink signature suffice? I know it sounds like a bit of a stupid question, but I don't want to get all the way there and then be told that my understanding wasn't correct.

By written they mean not just a verbal agreement (i.e. you need proof on paper), so either a typed-out or handwritten letter with their actual ink signature on the bottom.

2. What exactly would I need to show as "evidence that suitable living arrangements have been made?" Does that just mean I need a letter from my mother saying "Yes, I grant my daughter permission to travel from Boston to Manchester, etc. etc. etc. I understand that she will be staying with a friend, [insert name here], at the following address, [insert address here]. This individual can be contacted at [insert contact details here]." Or would I need further evidence from my friend's mother?

Normally, you would need written evidence from the person you are staying with in the UK, inviting you to stay with them, but it does say that your parents need to provide evidence about suitable living arrangements.

I would think a letter from your parents would meet this requirement, but if you can get something from your friend (or her mother) as well, inviting you to stay, that would be good too (maybe even just by email, since you're travelling so soon).