Right. And I think only (2) should count as local first software. And only (2) fits the definition provided by the local first software paper.
Option (1) is a legitimately useful way to build software. I wish more apps considered offline support. But I don't think "it works offline" is enough to make something count as local first software. For example, I don't consider Dropbox to be a "local first" tool.
I'm worried the "local first" term will be watered down by projects like this. That distinction matters. At least to me.