Is sundial time entirely dependent on solar azimuth?

Any sundial that gives the same result as this is correct and any other is wrong (but sometimes close enough):
_ /############
/| /#############
skewer / /############## (central) v / /###############
| north /################
| (S in S. /#################
| hmsphre) /##################
________|________ /###################
hoop | /##### LEVEL #######
(from the | side) /###### GROUND #######
| /####### (SOIL) #######
latitude--|- /#######################
(use || /########################
protractor)| /#########################
|V /##########################
| /###########################
|/############################
j#############################
,|#############################
/#|#############################
/##|#############################
/###|#############################
/####|#############################
/#####|#############################
/######|#############################
/#######|#############################
/########V#############################
/#######################################

#

write noon on hoop's inside closest to ground, midnight opposite, 6 pm on the east side, 9 pm midway between the last two, and so on (hours only occurring in darkness optional).

If your sundial reads 6 pm at due west all the time then you're doing it wrong. Let's say I put a vertical stick in the ground, draw a 24 hour clock face around it, put noon poleward and think it's a sundial. In New York City, it could literally be saying 6 am when a genuine sundial says 9 am. That's just middle latitudes. At the equator on the equinox, it would read 6 am all morning and 6pm all afternoon at the equator. If you go 1 mile south of where the next noon, summer solstice and Tropic of Cancer coincide, it would say about 4:30am at sunrise, go forwards at first, then backwards, finally showing the middle 6 hours of the night passing in 7 seconds. Backwards. At noon. Then it will run forwards again until it reaches 7:30pm at sunset. If you place the stick right, you can even make it stay between 4:30a and 6a all morning, stop at 6 am at the instant of noon then instantly become midnight, run infinity years per second backwards for an infinitely short amount of time, go almost 6 hours backwards in seconds, then later forwards again very slowly until it shows about 7:30pm at sunset. This is why sundials cannot be made that way.

(of course, this is theoretical, there are no infinitely thin, vertical, and straight sticks, shadows are fuzzy, they can be too short to see, the Earth wobbles a bit, the speed of light is not infinite, this would only be true if only the Sun and Earth existed, even a flea jumping in Russia moves the Earth etc.)

And yes, the sundial time can be up to 16 minutes away from mean solar time (the equation of time), easily noticeable, but if you wanted correct local clock time instead of correct sundial time then you could put as many dates as needed on another dial and rotate the hour scale until the arrows point at the current time of year. The shadow then shows mean solar time.

That should be close enough to mean solar time that you wouldn't care for a number of centuries, certainly a century if you're real picky. As for clock time, what sundial disagreements are possible is limited only by the whims of man. The sundial is several hours wrong in West China. Cause they use the zone that's good for Shanghai (or Tokyo when daylight savings).

And all sundials without moving parts are latitude specific. Some are adjustible, though. Some designs are more suited for some latitudes or even become impossible in some places, like the kind with a wedge or rod on a level face. They will also not work on days with polar night. Though you could use a moondial if it's also not polar moon's below the horizon for days or weeks. Yes, moondials exist! You need to correct for moon phase and time of year or they're useless.

Click edit to see the drawing. Don't click save of course.