How can I specify / format a contract role on my resume?

If you worked for a large company as a contractor (placed by an agency or otherwise), don't overthink it. Obviously, you want to anchor the resume screener on the bigger/more recognizable brand name of the parent company, not your dingy temp placement agency, so do just that. Here's what it should look like:

If you held multiple roles and titles at that one big company, you can also display that easily like I've shown above, but the second and third job title sections aren't necessary if you've only ever had one title there. (Deeper dive into showing multiple roles at the same company on your resume.)