Yes, with these boat load of agencies and their sometimes overlapping roles & job description, things can get confusing. I filed FOIA with USCIS using form G-639. What I got back was a massively comprehensive report compiled on a CD that included every bit of my 20 yrs immigr history records including first entry as F-1 student, my entry/exit records, H-1 filings, removal order/cancelation of removal order etc. and some sensitive/redacted info which I have no idea about. USCIS also forwarded my FOIA request to FBI which provided a separate 2 page report - it was not a standard FBI background check results, rather something specific to my immigr records.
From my expereince, USCIS is the mother of all, go to agency to get everything under one shop. So if you already have USCIS report, a separate FOIA to CBP or ICE may not add further value, but that's based on my own individual experience.