LION does not know about other websystems, but him just looked at a message sent to himself from a gail account, and him looked at the message source. To do this in RoundCube, there is a little down arrow at the far right of the abreviated message headder which will reveal the entire headder, and this information includes the fact that it was sent by a gmail account, AND the IP numbers that gmail received this information from.
At this level of network protocols, only IP numbers are used, However there should be a headder marked "Reply Path" which should have the senders name IF that name was provided to gmail (or hot mail) when that account was set up. That Line is: Return-Path: <broadwaylion@gmail.com> Obviously that line may be blank or forged or irrelevant such as for a user that wants replies to go to a different account.
The Line of interest is:
Received: from [192.168.42.69] (10G-N-Static-1-3.ctcinet.com. [173.241.180.3]) by smtp.googlemail.com with ESMTPSA id l5sm12717242oej.12.2015.10.06.10.11.53 for <elias@assumptionabbey.com> (version=TLSv1 cipher=ECDHE-RSA-RC4-SHA bits=128/128); Tue, 06 Oct 2015 10:11:54 -0700 (PDT)
Where the IP address 192... is my personal computer inside of our network, IP address 173... is our static IP address provided by our internet provider.
Now, I do not know if this information is preserved by your ISP provider when he forwards the message to you. Mail to our system is first sent to AppRiver who vets all incoming mail for spam. (We use AppRiver as our MX) If it passes their tests, they forward the messages to our server (hMailserver) for distribution to our users. RoundCube is on this server to provide web access to our users, and works directly with our mail server.
AppRiver preserves all incoming headers and includes them.
Both AppRiver and our servers preserve this information and include it in the received headders. Your ISP may not do this. It is part of the envelope that some protocol stacks strip away. It is only preserved if the servers handling the transaction preserve it in the message source. some apparently do not bother.
Elias