Yes, technically if could be considered downloading/uploading, but if you use it that way it may get confusing.
Downloading/uploading is generally used when you connect the computer to a distant source, for example a server on internet. If you use it for small devices, it's not as obvious which direction is up or down.
If you for example have a network storage unit in your network, it's not at all clear if it should be considered as a server or a storage device, and uploading would have completely different meaning depending on what definition you choose.
Although I agree with Brian Hitchcock that "portable electronic devices" is a much broader category than might be desirable for the range of devices that the OP is interested in describing, it (or its short form, portables) is the one that the computer magazines where I've worked have settled on to describe the relevant category.
The first distinction that the magazines had to make was between devices that were light enough to carry around ("notebook computers") and devices that weren't ("desktop computers"). But in practice, some people regularly transported fairly bulky electronic devices (to LAN parties, for example), so the issue of what was portable and what wasn't was always a bit unclear.
The next idea was that if a device could operate without being tethered to an electrical wall plug, it was a portable. But then a subcategory of large and bulky laptop computers appeared that were dubbed "desktop replacements"; they were not intended for everyday travel, although you could use them that way if you were willing to lug them around. Effectively, however, the laptop computer category itself became subdivided into portable and nonportable models.
Meanwhile, cell phones, PDAs, Gameboys, and mobile music players (Walkman devices for cassette tapes and then CDs, and finally MP3 players) coalesced into a subcategory of very portable devices dubbed "handhelds," the common characteristic being that you could carry them (and to some extent operate them) in one hand. But again new items—e-book readers and then tablets—appeared and muddied the waters, since they typically required a two-handed grip during extended use. So the broader category name for these items became "mobile devices."
This category name is complicated by the fact that it includes devices that connect to networks wirelessly and devices that do not, which invites disputes about whether the latter qualify as true mobile devices. The Wikipedia article on "Mobile device" takes the broad view that everything from a battery-operated calculator to a digital videocamera qualifies.
As certain types of laptop computers (notably ultraportables and netbooks) became smaller and/or lighter, they became more and more similar to devices in the "mobile devices" category. Indeed one type of product—the hybrid tablet/laptop was essentially a tablet (a mobile device) with an attachable keyboard that turned it into a laptop (a portable device). Nevertheless, perhaps out of habit, we never assigned laptops as a group to the "mobile devices" category.
And laptops aren't the only portable devices not generally categorized as "mobile devices." Also fitting into that broader category are such things as portable (wireless) printers and portable (wireless) scanners. At the other extreme, things like ear-attached Bluetooth devices, wireless headphones, and smartwatches are certainly "mobile devices" but (for obvious reasons) aren't classified as "handhelds").
The upshot of all this is that the clearest line that can be drawn between various electronic devices is between ones that can be carried easily by a user traveling from one place to another and that can be operated away from wall plugs, and ones that can't. Devices in the former group share the category name "portable electronic devices," or "portable devices," or "portables."
Best Answer
I think the lay (non-technical) person still thinks of "computer" as being a laptop or desktop. Of course, anything with a processor in it is technically a computer, which these days means your phone, gaming console, tablet, but also quite possibly your television, media device, and even devices like washing machines, microwave ovens and so on.
However, since you posted this question in the English stack exchange, I'm going to focus on how the general public understands the terminology, rather than technical definitions. I don't actually know of a single word which defines the concept of general-purpose computer, but that's what most people understand by "computer".
It is unfortunate that "PC" has come to mean a personal computer running Windows as its operating system, but that's the connotation in the public consciousness.
There's also "microcomputer" which once conveyed the same meaning, but it's become archaic since technically most servers now are also microcomputers.