('How' a reader of your query uses her own mind -- especially in any personal experience of creating the 'New' to some purpose -- will color all proposed answers, no?)
Some examples as come to mind, hardly exhaustive:

Psychiatrists.. there is truthiness in the koan re neurotics/psychotics who build imaginary castles/dreams and the psych-people collect the rent. In order to collect, they need to ~comprehend the structure in the subject's mind, in order to remodel towards a more practical daily existence for the client. (We can eschew the examinations of how effective are any of the Systems -- to this end.)

Walter Mitty.. and the quote which RFK likely lifted (maybe from Æschylus, too?) about making happen "things that never were". Anyone not-on-autopilot throughout life will inhabit this space at times.. and may very well connect such ruminations with her (livelihood) -- thus the practical role mentioned.

Linus Pauling, ill and abed -- virtually confirming the double-helix via a folded paper template with suitable "==" bonds-symbols he imagined to hook up.
(That he was later prevented from traveling to UK by the odious H.U.A.C. to report on his findings at a conference -- thus allowing Watson/Crick to claim the Grail … is another story.)
That was a New world, too, though not the metaphorical one of the programmer.

I recall, after grokking to medium-Fullness a melange of recently studied Boolean algebra tomes, seeing what a 'stack' + registers plus an arithmetic unit might be able to 'do' -- all re. the freshly unwrapped PDP-8 -- having an idea of what that mind-space might next be able to accomplish. This was even more fun when.. the hot new toy was, one day - a rilly-fast paper-tape reader, (found by a cohort from school-daze) also at lab, ordered and interfaced.

So, I believe that the 'programmer experience' is more common than your query supposes.
Just look at the layers of abstractions, starting from a search for 'human interface' quite beyond either flipping switches for a bootstrap paper-tape loader / or even the abstraction of ASR-33 mondo-typing.
(I'd bet that some non-tech folks dropped hints long before several ideas gelled, and assembly-language routines with vectors began to be cobbled-up.)
How does one parse the creativity march --> First [working] Mouse?

ie. The mind is a ceaseless generator of metaphor -- tailored to whatever consciously-directed problem it is directed to solve. Programmers may not, thus claim some Special Relationship™ to the engine(s) of 'Creativity'. IMhO.

(Managing 'creativity' --- now There's an oxymoron; never-mind that some folks wear such hats)
I believe that such folks [J.R. Oppenheimer being perhaps the Patron Saint of "herding cats" throughout the Manhattan Project] manage to 'manage' virtually via empathy with the local set of minds (individually), respect for the originality evidenced -- via the application of great tact -- thus, they 'direct' in the form of 'interesting suggestions'. Anywhere such a one begins to homogenize, treat all as an interchangeable 'set': is where the 'direction' will fail.

Just my experience in observing or participating in 'projects'. By the ModComp days, some of those projects indeed involved 'what programmers do' VS "how, inexactly? a massively complex accelerator is made to work Best." Lots of transliteration was involved in the above -- utterly inescapable where homo-sap is the End-user. (Much simpler if a missile aiming for Moscow is the End-user.)

In a word, every 'theoretician' employs skills comparable to programming -- it's a General human facility which has no associated job-classification (I can see.)

My 3 kop3ks

For the ephemeral nature of a 'mental world', nobody can impose 'constraints' upon artifacts a one creates;
only can they criticize the performance (towards stated ends) of the result -- in the world of pecuniary considerations.

As an Art (however disciplined by most of the koans of Science in such matters as 'speed of operation', accuracy and other matters amenable to measurement aka mensuration)
-- there is no Sole Right Way, as there are always n-ways via which some solution to a schema might be managed.
(How cute that this central-Fact protects from the dogmatic influence of the religionists, as sometimes impose egoistic Will-fulness upon so many other paid activities.)
And yet.. the PHB shall always be with us, it seems; well.. so long as the environment is of the current bizarre-form, vulture-capitalism, that is.

As a programmer does not consume tangible materials, she is free to create Any structure out of Any internal map which, in the end -- adequately accomplishes the general specs.
Freed thus of engineering costs and materials fabrication ... why Yes, very few can claim possession of as much near-complete-autonomy as the non-regimented Programmer:
at least, after having established her chops, thence ascending (along with the quality of a succession of results): towards --> Gigundo Omnipotent Designer (eh?)

(Still.. one Marc Newsom -- a kind of general Designer of many kinds of objects, interviewed on Charlie Rose on 10/1 -- appears to possess also that degree of laissez-faire)
... based also on previous successes, sufficient to demand similar autonomy (although he acknowledges certain constraints),
as where a certain company's 'style', for want of a more precise description, is to be incorporated along with The New.
In that regard, it still looks as if the Mondo Programmer remains at the top of the heap re. 'freedom' !!
-- in a world perpetually obsessed with much-Much-MORE than "The Adequate" -- well, for as long as That world lasts.

2 kopeks more.