To avoid the thread becoming too confusing:
@live.com is referring to a online live MS account. It can be of several flavours. eg. Outlook.com
Outlook.com is the new/current name for the Microsoft online Email service. If an account is opened there, the "outlook.com" would be the suffix.eg (e-mail address removed). If you have no real need for a desktop program, this is an excellent alternative.
Live Mail (which I use) is a desktop email program, which can fetch online Email automatically to your desktop. AS Trouble says, it is part of Live Essentials. It is a slightly slimmed and simplified version of MS Outlook.com. It no longer receives official help from MS as it's production has been discontinued, but still works 100 % on all platforms.
Ms Outlook.com, is party of MSOffice. It is a very reliable program, with functions over and beyond simply handling mail. Comment here would be that, if you have no real need for the MS office program, with its cost and used HD space, it is an inefficient way to get Email.
"Mail" is the new, over simplified, mail program in Windows 10. It is not receiving too much custom, for the moment, but , according to MS, is still being developed and improved. OIt's lack of functions may be one reason for the lack of enthusiasm.