What are a coolie and a draw, as in "feed in the coolies & water in the draw"?

Coolies

The more common spelling of the first word is coulee (though cooley and coolie are also both found). It is a North American dialect term for a deep ravine (originally ‘lava flow’—a somewhat unusual semantic development). The OED has a slightly more detailed definition:

“In the Western regions of Canada and the United States: A deep ravine or gulch scooped out by heavy rain or melting snow, but dry in summer.”

According to them, “Sense 2 [the ravine, rather than the lava flow] appears to have arisen among the French trappers in the Oregon region”, though their citations for the ravine sense predate their citations for the lava sense by about 30 years.

Draw

In this sense, a draw (look under the noun section, sense 6) is –

“A gully shallower than a ravine” (Merriam-Webster)
“A natural ditch or drain that draws the water off a piece of land. Also, a shallow valley containing a stream.” (OED)

As far as I know, neither word is in particularly common widespread usage (I’ve never personally heard draw used in this sense, though the comments indicate that it is more common than I thought), though I expect they may be in certain particular areas or vocations.

From my own experience, growing up in the US mountain west: "draw" is pretty common, at least as much so as "ravine." I'd define it as a drainage smaller than a canyon.

"Coulee" is very common in the Northwest US, particularly Washington state (cf. the Grand Coulee, as someone mentioned), and in fact, having lived in Washington for a number of years, I think of it as a northwest localism. It tends to be used instead of "canyon," particularly of a flat-bottomed canyon with steep sides. As used there at any rate, a coulee is not a small or shallow feature!

Coulees are larger, deeper, and have steeper sides than draws. The Grand Coulee Dam was built at the narrowest point between ravine/draw/coulee walls.

Appropriate as the word grand is for the dam, the name Grand Coulee actually refers to a unique geological feature created thousands of years ago when Ice Age glaciers blocked the ancient riverbed of the Columbia River. When water eventually traveled under the ice dam, great floods scoured out hundreds of miles of deep gulches, or coulees, in the Columbia River Basin. One of these, the Grand Coulee, is four miles wide and bordered on three sides by steep cliffs.
Washington: Grand Coulee Dam

Is the setting in the Dakotas? This sounds like the typical Dakotas' way of describing the land. (I've spent time in all 50 US states, and I lived in the Dakotas for seven years.)

Here's a description of Munch's Coulee near Kenmare. Note that the description also refers to a draw.

Munch's Coulee Hiking Trail is located approximately 5.5 miles south of Kenmare on the Des Lacs National Wildlife Refuge Auto tour route. The mile-long trail includes a quarter mile that is level, hard-surfaced and easily accessible for wheelchair users. The rest of the trail is grassed and is steep in places, but provides a spectacular view of the valley, marshes, and wildlife area.

The bottom portion of the trail is in the draw where excellent bird watching opportunities abound. There is a small pool of water within comfortable viewing distance from the first bench. Turtles and duck broods can regularly be found in the spring. The trail then climbs up the side of the draw to the upland prairie

https://www.americantrails.org/resources/munchs-coulee-trail-kenmare-north-dakota

To me, a draw is a side canyon where one expects to find a water source. If you want clean water, you head up a draw to a source. Livestock tanks are typically nearby. (A tank is any reliable water source for animals, usually improved in some way to be more reliable or better accommodate groups of animals.)

.

The meanings of 'coolie/coulee' in American regional English

Mitford Matthew, A Dictionary of Americanisms on Historical Principles (1951) offers a specific and seemingly straightforward definition of coulee:

coulee, n. A small stream, or the bed of a such a stream when dry.

But according to the Dictionary of American Regional English (1985) a coulee can refer to four distinct types of watercourses or topological features depending on the part of the United States where the term is used. Here are the relevant parts of DARE's entry for the term:

coulee n Also sp cooley, coolie, coolly, coulé(e), couley, coulie {Fr coulée a flow; ...} 1 Any of var types of watercourse: a A stream bed, often one that runs dry according to the season; also the stream itself scattered, but esp L[ouisiana] [Citations omitted.] b A small bayou or canal in marshland. esp L[ouisiana] [Citations omitted.] A valley or depression between hills. chiefly W[isconsin], M[innesota], M[ontana], [N[orth] D[akoya], W[yoming] [Citations omitted.] d A ravine or valley with sloping sides. chiefly West [Citations omitted.] ...

The last of DARE's citations for coulee in sense 1(d) of the word is from Ramon Adams, Western Words: A Dictionary of the American West (1968), which provides the following definition:

coulee A dry creek; a deep cut in the earth's surface, its sloping sides covered with brush; a ravine;used in the Northwest as a synonym for the Southwest's arroyo; from the French coulée, meaning flow.

You might suppose that a cowboy song like "Old Paint" would have Adams's definition—and DARE's definition 1(d)—in mind in its use of coolies, and you might well be correct. But there is clearly some overlap in Western U.S. usage, as Montana and Wyoming qualify as Western states under any reasonable definition, and North Dakota and Minnesota are arguably Western (or at least northwestern Midwestern), too. Meanwhile, Louisiana is on the border of cowboy country itself as it shares its western border with Texas.

So while the "ravine or valley with sloping sides" meaning is probably the one intended in the song, the "stream bed ... that runs dry according to the season" and the "valley or depression between hills" meanings are not out of the question.

---

The meanings of 'draw' in American regional English

Adams, Western Words (1968) has this entry for draw:

draw A shallow natural drain for rainfall. ...

And Matthews, A Dictionary of Americanisms (1951) confirms this meaning:

draw, n. ... 5. A natural drain or gully, a coulee, a ravine.

But once again, the Dictionary of American Regional English (1991) complicates the picture with multiple distinct geographical meanings of the term:

draw n ... 2 Any of various types of of valleys or low places; spec: see below. Note: It is sometimes difficult to distinguish among the following senses. a A steep-sided gully or ravine; a stream-bed, often one that runs dry. chiefly West Cf arroyo [Citations omitted.] b also drawer: a gently sloping valley; a dell. chiefly West of the Missi[ssippi] R[iver] Cf dingle [Citations omitted.] c A passageway through a mountainous area. Cf gap, notch [Citations from Wyoming, New Hampshire, Massachusetts, Wisconsin, Colorado, Illinois, Indiana, Kentucky, Missouri, and New Mexico omitted.] 3 A small stream or stretch of still water Cf draft [Citations from Missouri, Illinois, Virginia, Texas, and Tennessee omitted.]

---

The meanings of 'coolies/coulees' and 'draw' in 'Old Paint'

At first blush, it might seem that the lyrics to "Old Paint" use both coulee and draw to the refer to the same geophysical feature—essentially, a ravine. That is, "they feed in the coolies and water in the draw" could be taken to mean "they feed in the ravine and water in the ravine"

However, coulee and draw have multiple specific meanings that are not entirely interchangeable, and it may be that the lyricist instead wants listeners to interpret "coolies" as referring to the banks of a ravine and "the draw" as referring to the bottom of it, where water accumulates and stands until an extended period of rainless weather dries it out. Another possibility is that the lyricist imagines individual coulees as being smaller ravines that branch off from the main watercourse ("the draw"), which itself has the features of a large ravine.

For its part, DARE seems to invite listeners to choose from an abundance of meanings, each depicting a different scene of cattle scattered across a varied landscape where the animals find suitable places to eat and drink at their ease.