I heard the first, but not the second. This is advice. There are no rules regarding
H1 tags. Search engines cannot dictate how to use HTML. Use as many as you want however you want.
However, the advice to use just one
H1 tag is generally a good one and fits convention. It is strong SEO advice. In addition, the
H1 tag should strongly support the title which is also supported by a more verbose rendition as a
description meta-tag. This is standard advice for a good reason. It works. As well, following convention will not trigger any red flags.
With so many header tags available, there are plenty of options as not to need more than one
H1 tag. With CSS, style is not an issue. I am not sure why someone would use more than one
H1 tag under normal circumstances. I am not saying there may not be a valid reason to, just that I cannot imagine one.