In the world of water science, “rivers” and “streams” are used almost interchangeably, but “stream” has more currency. All linear waterways are streams. But “river” tends to be reserved for larger, boatable waterways. Still, they are pretty much the same thing.
Someone should film a chase scene in that for a movie. That would be cool. 
To me, a stream is the smaller bit of flowing water compared to a creek or brook.
In my dialect (whatever it is), stream (specific to water) means any flowing water. The water running out of the faucet in the kitchen is a stream of water. The water coming out of the garden hose is also. A river is a stream of water too, whatever size. So’s a creek. A current in the ocean is also a stream of water. (The Gulf Stream, anyone?) Water streams from my hair when I turn off the water in the shower.
So, I can see why “stream” wouldn’t be common in the name of rivers or creeks or brooks - those three words seem much more specific. I’d never apply them to water out of a garden hose, or an ocean sized current, either one.
To me a stream has always been a small creek, with a creek being a small river.
Approximations:
Stream: between a trickle and about 1 foot wide
Creek: between about 1 foot wide and about 8 feet wide
River: 8+ feet wide
To me the words “brook” and “stream” mean the same thing, but I don’t use “brook”.
For me, there’s really no size distinction between a “creek” and a “stream.” They’re synonymous. If forced to classify them, I would instinctively say that a stream is smaller than a creek. And a brook to me is the smallest. So, for me, it would be brook > stream > creek > river, except that I use “stream” far more generically than that.
This is how I understand it; stream is the general term and brook, creek, river, etc. may be used depending on the size, but there are no rules.
BTW, in upstate New York, they use the word “kill” for rivers and creeks. As in Beaverkill River.
It’s statewide. Everyone in NYC and northern NJ has heard of the Arthur Kill.
To me a stream is a shallow creek. I know that’s odd but …
In a watershed rivers often feed into other rivers as well.
So is “creek” to mean a tidal inlet peculiar to the UK? I can’t think of any freshwater streams that are known as creeks over here.
A tidal inlet’s an estuary to me.
A stream is any river, I’d say “let’s go walk by the stream” when I mean what’s officially the Liesbeek River, for instance. It’s not a classificatin to me, it’s more like a shorthand for “the flowing water, you know the one I mean, it’s obvious by context”
Nor me. ‘Creek’, IMO, refers to the watercourse(s) within an estuary - not necessarily including the mudflats, marshes, etc.
Some creeks (for example those in deep, narrow, rocky valleys may not seem particularly estuarine in general nature - they can be just a narrow inlet with sloping rocky sides.
Nitpick: mostly or exclusively in the Dutch-settled portions of NYS, i.e. downstate, NYC, and the Hudson Valley, where the Dutch derived “kill” placenames took hold. I don’t know of any kills in Central or Western NY although there may be some.
My favorite is the inappropriate-sounding waterway north of Schenectady called Indian Kill.
As long as you don’t pronounce it “crick,” I’m fine.
We have a rather large estuary in my area that goes by the term “creek”. Now it makes sense, considering I live in New England 
Usual terminology of creek is what I’d consider to be a brook in the southern US.
This is an interesting topic to me as I spend a lot of time chatting with another naturalist friend who lives on the opposite end of my mountain range in PA. He has eels and crayfish in his “creeks” which appear to contain class 4 or 5 rapids among the boulders. I have snappers and “crawdads” and my creeks are wide, shallow, crystal clear flat water wading pools with a few shoals here and there. Not wide enough to float in, not fast enough to drown in, and not wild enough to bother putting a boat on.
'Round these parts are one of the last strongholds of brook trout, which thrive in hemlock shaded mountain brooks and streams, narrow with the occasional deep pools and swirling foam around a few scattered and well-worn boulders. (Jefferson and Cherokee National Forest)
I have always used streams to be smaller than a creek, which is see is the opposite of what most posters think.
A rivulet would be smaller than 12 inches.
A stream would be between 12-36 inches.
A creek would be 3 feet to 12 feet.
A river is bigger than that.
That pretty much sums it up.
Of course, in Illinois, sometimes it was a “creek” and sometimes it was a “crick”…not quite sure why both pronunciations were used, but people seemed pretty rigid in describing certain specific local waterways as a “crick” and others as a “creek”. Afraid I can’t help in determining the differences between those two.
In New Mexico, the label “creek” is not common, but the few places identified as “creeks” are as substantial as any river in the area (and most of our rivers would barely count as creeks in a place like Virginia or Alabama). Carrizo and Rita Blanca Creek are arguably far more substantial than the Rio Ruidoso.
Other names for waterways of this size are arroyo or gulch (if dry most of the time, except after a heavy rain) and draw (basically any shift in elevation where runoff may gather or channel). West Texas is full of “draws” with colorful names and bridges - but rarely any water.
As a fellow South African, I’d say that I do classify by size. There’s a large overlap in the middle of things that could reasonably be called “stream” or “river” - the Liesbeeck being in that overlap. But there are definitely some waterways that are too big to be referred to as a “stream”, like, say, the Breede River at Swellendam. (And conversely there are many small streams in the mountains that I’d never call a “river”.)
Also, there are some things named “X River” that I would definitely consider to be streams at some point in their course, and possibly vice versa as well.
(I was originally going to say that I consider a stream something you can jump across, or at least consider jumping across, but actually that’s too restrictive a constraint.)