I ain't no IT expert but it wouldn't involve your computer all that much. It would involve the server IP logs, and the ISP logs. Most all of us have dynamic IP addresses, which means every time we shut down our computer or reboot our ISP gives us a fresh IP address (within a group of IP addresses that they own).
So if server logs showed connections from certain ip addresses, the suspect IP addresses would be compared to the originating ISP logs (your internet provider keeps logs of who used what ip at what time) and that would identify who specifically connected to the server.
The actual home computer might show websites visited, and cached pages that were visited, and of course auto login info for those sites, but actually IP info? Nope.
For instance if the Feds were running a fake child




site or whatever, they would log every IP address that connects to that server, then they would look up IP number group and get a warrant for the ISP that owns those numbers so they could check the IP logs and therefore they would know WHO specifically used that IP address at that time specific time to connect to their fake child




site.
In simple language, it wouldn't be the home computer vs the website server, it would be the website server logs compared to the internet service provider logs (not likely either are faked, really really not likely).