No. You need to distinguish between the different types of services, such as recording vs. [live] viewing. Those are two different actions. Other actions would be streaming, pay-per-view, on-demand, renting, buying, etc. It's any action you could take with content on your TV: view it, stream it, record it, buy it, etc.
You also need to distinguish between the different types of access (connection): cable TV vs. Internet.
Recording a show used to mean recording a show to a local storage device, such as the hard drive of a DVR or computer...or a VHS tape. All these are local storage. However, local storage has vulnerabilities. Your equipment could crash and destroy your recordings or a VHS tape may eventually wear out.
To overcome these vulnerabilities, providers now offer servers to store your recordings outside your house (into "the cloud"). However, I'd use the term "cloud" very loosely.
"Cloud" assumes Internet. Internet assumes accessing content through your Internet modem. You don't need an Internet modem to view TV shows. You only need a cable box (or antenna) and a cable box is attached to the cable-TV distribution system. Internet is a different connection.
Yes, this new cable box may store your recording into "the cloud" but it doesn't mean you'd access it via the cloud (Internet modem). I'd assume you'd access these recordings via the cable-TV distribution system. Is there a difference? Yes: data caps. Internet service is metered for data caps; however, you could watch cable TV 24/365 without data caps.
So cloud does not mean "...viewing via Internet on computer." That's streaming. Streaming is solely accessing content via a metered Internet connection.
For example, if you recorded the Super Bowl, it'd be nice if you could preserve that recording...so record it to the cloud. Although it's saved in the cloud, it'd be nice if you could view it sans data cap...so access it via the unmetered cable-TV system. Although it was recorded via the cable-TV system, it'd be nice to view it on my computer...so access it via the Internet and stream it against your data cap.
"Cloud" is a loose term when combined with cable TV, but there are 2 takeaways for you: where you record a show and how you access the recorded show.
Although Cox is marketing their DVR as "cloud" storage, it's stored on the cable-TV system.