Cfxtrjtrk

draw (or re-entrant)

Draw (terrain)

A draw differs from a valley or an arroyo, in that the ground always
slopes downward from a draw in only one direction, and upward in the
other three.

Wikipedia

Picture from armystudyguide.com

To be more specific than valley, it's a combe.
Alternative forms are "comb", "coomb" and "coombe".

It defines a short valley or deep hollow, esp in chalk areas, a valley
enclosed on all but one side.

source

Inlet is used only for water-filled valleys, so wouldn't be appropriate. Valley is correct to describe these formations. Most valleys do not go 'all the way across' a mountain. That would be called a 'pass'

Worldwide there are lots of local words for this type of thing. One of my favourites is cwm, (pronounced "koom") originally a Welsh word but accepted in English, meaning a small bowl-shaped valley. You have already mentioned coulee, which is more general and could be applied.

Another English term for something similar, but specifically with steep sides is corrie (from the Gaelic coire). They can also be referred to as 'bowls'.

(Incidentally, the definition of cwm is evidence of the use of valley for a formation that doesn't go all the way across a mountain.)

Notches or Saddles.

As you can see on Bald Mountain:

A series of notches can create a saddle.

gulch

: a small, narrow valley with steep sides
M-W

cove

: a deep recess or small valley in the side of a mountain
M-W

I would call such a feature simply a valley. In contrast, a passage going through the mountains, across a saddle point between two or more peaks, is what I'd call a pass.

Note that even valleys formed by rivers or glaciers — which, by their nature, inevitably slope in one direction, away from the mountains — can be quite long, and surrounded by multiple distinct peaks and saddles. Thus, it's quite possible for there to be several relatively low and easily passable ways in and out of a valley — even if, typically, there's only one way out of a valley that doesn't require you to go uphill at all. If I wished to emphasize that a particular valley was completely surrounded by mountains on all sides but one, with just a single easy way in and out, I might describe it as enclosed, or as a cul-de-sac, or indeed simply as "surrounded by mountains on all sides but one".

(And no, whether the valley is "U-shaped" or "V-shaped" has nothing to do with it. Those terms just describe the cross section of the valley, glacial valleys being typically U-shaped, while river valleys are often V-shaped.)

In the southeast U.S. these are called "hollers", which I always thought was a heavily accented pronunciation of "hollows". Or maybe it's because you can shout across one holler, but not from one into another.

An example of usage might be, "I think Jim keeps his still up the holler." Or if you are hiking with a bunch of kids that keep asking how far it is until camp, the stock answer might be "just around the next holler" (of course, this never comes true).

Dale is roughly synonymous to valley, but has a softer connotation I think. See Wiktionary.org:

dale ‎(plural dales)
1. (Britain) a valley in an otherwise hilly area.
"Through wood and dale the sacred river ran," - Kubla Khan by Samuel Taylor Coleridge

How about canyon? And, if it goes nowhere, box canyon.

Colloquially these are often called crags or clefts.

A lot of this is very colloquial and the differences between them if any, are very location specific. In the northwestern U.S., the word I most often heard was:

Gorge: (1) a narrow cleft with steep, rocky walls, especially one through which a stream runs. (2) a small canyon. (Source)

Synonyms: ravine, canyon, gully, pass, defile, couloir, deep narrow valley; chasm, abyss, gulf;
dialectchine, bunny;
clough, gill, thrutch;
cleuch, heugh;
gulch, coulee, flume;
arroyo, barranca, quebrada;
nullah, khud;
sloot, kloof, donga;
rare khor

I suspect most of the problem is that there are a bazillion borrowed words that refer to that small space between two ridges that, themselves, lead to the peak of a single mountain. I was most entertained by the rare synonym "khor," which apparently originated in the Sudan (not usually the place to inspire images of tall mountains) as a "dry watercourse or ravine."

In my locale, a "valley" is something wide and often (usually) the lowest region between two mountain ranges, not simply between the spurs of the same mountain or the pass between two mountains of the same range. However, I wouldn't doubt it's used all the time to refer to any or all separations between two ridges (mountains or not).

Depending on the mountains (and location), the inlets could be considered fjords.

Another word that could work (depending on the imagery you wish to evoke) is slope. In a general sense, that could describe any of the sides of the mountain.