Yeah, one of my main priorities for a home server is its energy efficiency (and fan noise). Older laptops rarely fit into that. But newer 'ultrabooks' might be good.
Laptops will typically always use less power than a desktop.
But desktops can be cooled much more silently than laptops.
And performance will almost always be better in a desktop. But might also be overkill for what you want to do.
Personally, my 2 servers are a Raspberry Pi 4 running off a USB SSD, and a Synology DS220+, where I found an 8 GB stick of RAM that it would boot with for a total of 10 G.
This perfectly suits all I want to have at this time, with a combined continuous power draw of around 15W.