That's a cloud-centric interpretation. Like using CDNs. That'e been around for a while.
What I think will be interesting is intelligent processing and storage on end-node devices, like a home gateway, smart appliances, or wearable devices.
Instead of sending the data to the cloud for calculation/analytics, it does it right there on the device.
For example, an Alexa or Google Home device sends everything you say after a wake-word back to Amazon or Google. A device with sufficient edge storage and compute would be able to do the same without sending your voice outside your home.
We're not quite there yet, but it's getting closer.
I'm pretty sure that's not what edge computing is. You've just described client-side computing.
The "edge" is similar to a CDN. Usually some kind of application layer code that's running in an ISP data center rather than in a cloud provider's data center.