Breaking your overall question down you ask:
Would someone with darkvision or Devil's Sight be able to discern those lighting differences we consider important to our normal vision?
Would someone with darkvision or Devil's Sight be able to detect where subtle shadows are when there's a faint light source to cast them?
Do the rules have anything to say, or is there guidance from closely connected material or other editions available?
A discussion of the difference between these the rules describing darkvision can be found here what is the correct interpretation of darkvision and the rest of this answer will assume that darkvision is consistently applied as written in the race descriptions (e.g. dwarf PHB p.20) though it actually does not make much difference to the answers.
Darkvision PHB p20.:
You can see in dim light within 60 feet of you as if it were bright light, and in darkness as if it were dim light. You can’t discern color in darkness, only shades of gray.
There are two specific levels of light covered here:
- Dim light, where there is a source of light
- Darkness, where there is no source of light
In both cases darkvision enhances vision such that "you see as if it were [the next higher level of lighting condition]". Obviously you can also see at all light levels in between using the same "as if" rule and there can be no shadows if there is no light source. As always the light level present are adjudged by the DM.
Addressing your questions one by one:
1. Would someone with darkvision or Devil's Sight be able to discern those lighting differences we consider important to our normal vision?
Using the "as if" rule YES, with the exception that colour is replaced with shades of grey in darkness. If a creature with normal vision would be able to discern the lighting difference in the "as if" lighting condition, then so can a creature with darkvision in the actual lighting condition. Otherwise they can't.
2. Would someone with darkvision or Devil's Sight be able to detect where subtle shadows are when there's a faint light source to cast them?
The question appears to be asking "would you be able to see subtle shadows if the lighting conditions were less than dim but more than darkness?"
If the DM adjudges that a creature with normal vision in the "as if" light level would be able to see the subtle shadows in question then YES, a creature with darkvision can see them in the actual lighting conditions. There is obviously no sudden jump between darkvision's benefits in dim light to those in darkness, it is continuous.
3. Do the rules have anything to say, or is there guidance from closely connected material or other editions available?
The RAW from the 5e core rule books actually cover the questions you ask. I can find no other guidance specific to your questions in 5e supplementary material. As stated elsewhere, rules from other editions are not a reliable guide.
To list the sources defining darkvision that I have found in the core rule books:
- PHB p183 Darkvision
- PHB race darkvision descriptions (e.g. p.20 dwarf)
- Monster Manual p9 "monster" specific darkvision
And there is, as referenced above, what is the correct interpretation of darkvision
