Nice. Hearing back is about 50-50, or lower, in my experience.
Well, I have to say this: if one tree on Myheritage has Lars Nilsson -> Nils Nilsson, then there WILL be several trees with the same info. Falsities spread as easily as truths.
Cadastre? I had to look that one up. I guess it is what I call the taxation records. They are available at Riksarkivet SVAR from 1642 and on, covering more or less every year (and Piteå does not seem to be a "gappy" parish, so that's good). The usual format is a list of the heads of household, with columns for tickmarking the members of the household: head,wife, number of sons over 16, number of daughters over 16, sons-in-law, daughters-in-law, and possibly a few more. No names for anyone but the head of household, no birth dates. (If you are lucky there may be more informative variants, but that will be later in time and far from all parishes).
The exact birth date given for Lars Nilsson probably derives from the birth book I described above. He is not in the first household record, so it cannot be from there.
When you research the taxation records, you basically have to map the households in the village, and draw conclusions from the differences between the years. I suppose/hope somebody has done that.