I think it would be useful to add this quote to this thread, because it partly explains why people ever started bandying around the word 'sloppy' in connection with his playing:
“Terrible. Really sloppy. I'm just totally uneducated. An illiterate guitarist, really. But it doesn't make any difference because now and then something good will come through.”—Jimmy Page defining himself as a guitarist, from Melody Maker Sept. 1974
When Page said this he was being self-deprecating to some extent (he is English, after all). That is an exaggeratedly negative self-assessment.
As has been said earlier, the imprecisions and limitations of his style were hardly unusual in the time-frame when his style was formed - i.e. the mid-60s to early 70s. Rock guitar was considerably less technical back then. But they are an important part of his style. I think most rock guitarists would give their right arm to be 'sloppy' if it meant they came up with the solos and riffs which Page came up with.
Some of what people might perceive as 'sloppy' is actually timing. Page's classic solos often play 'late' on the beat. I've been a professional guitar teacher for 35 years and everyone whoever came to me and played the studio Stairway to Heaven solo never got it right even when they had all the notes in the right place - why? Because they never get his late timing. They play it right on the beat, with a kind of 'oh this is easy' attitude. To really play it you've got to make it sound like its an almost impossible struggle to get the notes out - a good example would be the solo on 'I'm Gonna Crawl' where he really weights a lot of the phrases by playing late. It sounds fantastic, of course.
The earlier observations about the change from wrist to 'scrubbing' are very true.
Page's tragedy as a musician is that he could not escape the persona he created for himself in the mid-70s - the problem was, who would want to? But then the band ended, and he could not find a vehicle in which to place that persona. Jimmy Page was the person who never left Led Zep. Jones and Plant did not allow their musician impulse and creativity to be so restricted by a persona, hence the diversity of their solo careers. Page has always been reluctant to do anything musically which might be perceived as 'not very Jimmy Page' as defined by that persona. Hence the archivist role, and the attempts to play Zep songs again in various formats and bands.
Sorry this has gone on a bit ...