JPJ has said numerous times that he used Roto rounds. Just read some of his interviews. Where is the source showing he used flats? Anyway, JPJ's tone is nothing special in early Zep records.
So, which albums are the ones where his tone IS special?
Also, to the OP, don't assume that he necessarily used the Acoustic 360 in the studio on LZ II. This album was recorded for the most part in bits and pieces at the time of their 1969 US tours, and as can be seen here...
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...their preferred backline didn't always show up in time for live dates, as the concept of equipment logistics for rock band tours was still in its infancy. It would appear that JPJ didn't even take possession of the Acoustic until around mid-'69; the earliest US tour shows him using a combination of Rickenbacker Transonics, Marshalls, Sound City/Hiwatts, and in a pinch, borrowing Ampeg and Sunn gear. Their tour gear might not have even gotten to a studio anyway, forcing him to use what was on hand. Their first ever tour, in Sweden, shows them using Vox amps, probable Yardbirds hand-me-downs, and one of those might be the likely candidate for what was used by Jones on the first album, but while Jimmy Page has been willfully vague on his equipment choices at that time, JPJ hasn't exactly been forthright either, so who knows?

