? I always thought 1800s meant the first decade of the 19th century rather than the whole century, so that we went 1800s, 1810s, 1820s, 1830s etc.
I would take the meaning of "the 1800's" according to context. If someone made a sweeping statement such as,
"the 1800's saw many societal changes, the primary one being the industrial revolution", I would take it that they meant the entire century. Perhaps this is a cultural difference? I have noticed, now that I think about it, that some British folk seem to put 19C to refer to the whole century. I don't think that's common in the U.S.