How to judge whether my Xbox360 is running a genuine system

Check to see if all the labels and rubber feet on the console are intact. If missing, the Xbox may have been opened up at one point, voiding the warranty. This doesn't necessarily mean it has been modded though. Unless you dive into the system files of the console, then you probably wont be able to tell.

It`s also advisable to do a full factory reset and update of the console before you download your profile onto it.
I`ve been selling Xbox 360 consoles for a few years now and always make sure the factory seal is intact before I buy or sell.

Wasn't enforcement united focussed on the offensiveness of gamertags? How can regular users possibly have the tools to judge whether someone cheated or not?

It's a weird thing but you are genuinely one of the bars I use on whether to buy a game or not, Az. In that the more you like a game, the more likely it is that I wont.

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Not the best system perhaps but it works for me.

Whereas it is a bummer that the family plan is now gone, it is what it is. You will only need two Live accounts for your set up.

Xbox One = One Live account = anyone who uses this system will have Live through the person associated with the console.

Xbox360 = One Live account = the same account tied to the One.

Xbox360 = Second Live account = anyone who uses this system will have Live through the person associated with the console.

Xbox360 = Second Live account = The person with the second account uses this system.

OOORRRR... Have all systems tied to the same person. Instead of different people. Anyone who uses the system connected to a Live account has Live through them. Whether that person is signed in or not.

Example:

I have Live. I have an Xbox in my bedroom. Anyone who goes into my room and uses my Xbox will have access to Live whether I am signed in or not.