Thanks both! I think I see where the confusion lies. Both Comet I 1862 and Comet II 1862 where discovered in the same month (July 2nd and July 22nd). I’m not sure how quickly discoveries became known to amateurs at that time but Norman reports very rapid motion of the comet, which suggests it was Schmidt C/1862 N1 which was making a close approach to Earth also known as Comet I 1862. This reference here: https://www.sciencephoto.com/media/364340/view/comet-of-1862 observed by Thomas William Webb which looks very similar to Lockyer’s drawing of around the same date was labelled as Swift-Tuttle also discovered in July of that year! Must have been a vintage year for bright comets!
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