Lightroom Classic always keeps originals on local storage. There is no option to store a catalog or originals in the cloud. Lightroom Classic can sync Smart Previews to the cloud, but not originals. There is no option to sync edits across multiple computers running Classic, partly because the cloud storage can sync with only one Classic catalog at a time.
Cloud-based Lightroom always uses cloud storage for originals, so if that version has the features you need, and you can afford the rate for the cloud storage capacity you need, that is the only way to keep edit originals on multiple computers seamlessly, and it requires that originals be in the cloud too. (For better performance, it is possible to have cloud Lightroom keep local copies of originals as space allows. But the actual originals must always be in the cloud.)
If you must use Classic for the features and workflow, and across multiple computers, what many people do is put the originals and the primary catalog on a large external volume (for example, a high capacity bus-powered SSD) carried to and connected to the computer you want to edit on. This also requires keeping Lightroom Classic configuration settings (preferences, presets, keywords, autofill lists…) consistent across those computers, which you have to set up yourself (such as with syncing software) because Lightroom Classic does not provide an automatic way to do it.