I grew up calling these "daddy longlegs" - a name also applied to cellar spiders. But it was harvestmen we always saw (and tried to avoid stepping on), and which we still see on a routine basis when gardening or working in the woods.
They are not spiders. They are arachnids, but are of a different order from true spiders.
You learn something every day.
Addendum: And this is cool...
The legs continue to twitch after they are detached. This is because there are 'pacemakers' located in the ends of the first long segment (femur) of their legs. These pacemakers send signals via the nerves to the muscles to extend the leg and then the leg relaxes between signals. While some harvestman's legs will twitch for a minute, other kinds have been recorded to twitch for up to an hour. The twitching has been hypothesized as a means to keep the attention of a predator while the harvestman escapes.