According to Wiki they also say the two answers you quote claim to be the highest point. However, they say they are the highest point in the ceremonial county of Lancashire but, The highest point within the historic boundaries is Coniston Old Man in the Lake District at 803 m (2,634 ft).
It goes on to say "The Old Man is the highest point in historical Lancashire, though following the 1974 revision of county boundaries in England it now lies, for administrative purposes, within Cumbria".
Unfortunately you will have to take your pick.