Although I am probably not able to identify the names/terms of the chord variations, my early songwriting already used the following: Major7, Major Add9, Minor7, Minor6, Minor Add9, Aug4, 7 Sus4, 7 Add9, Major9. They are like people whose faces we recognise, but whose names we don't remember.
Examples in C and Am minor key signatures, and in ascending order of chords:
C Maj7 = C, E, G, B.
C Maj Add9 = C, E, G, D.
A Min7 = A, C, E, G.
A Min6 = A, C, E, F#.
A Min Add9 = A, B, C, E.
C Aug4 = C, F#, G; or C, E, F#, G.
(Fifth chord) G7 Sus4 = G, C, D, F.
(Or if I'm going to move from Am to D7, I'll use D7 Sus4) D7 Sus4 = D, F#, G, A, C.
(Another variation of D7, following Am) D9 = D, F#, A, C, E.
(Fifth chord) G9 = G, B, D, F, A.
The variations in the chords (as opposed to your typical orthodox 1-3-5's/C-E-G's) allow more options for stringing chords together since the virtual counterline of notes that connect one chord with the next will be tighter. At least that was my take on it (and at least it explains why Canon in D "flows").
For instance, in the key of C major, let's use the example of C chord transitioning to F.
The constituent notes of C major are C, E and G, while F major is F, A and C.
There are a handful of virtual counterlines that would connect these 2 chords:
1. C to C (0 key)
2. E to F (1 key)
3. G to F (2 keys)
4. G to A (2 keys)
For me, it is not the individual, but the collective set of counterline transitions that determine the extent to which one chord will "flow" to another chord. I religiously apply this principle to my songwriting because I always believe it makes the most sense for the style of guitar-driven pop-rock songs I like to make.
Now, when I "complicate" and "fuzzy up" the chords C and F to be, say Cmaj7 (C, E, G, B) and Fmaj add9 (F, A, C, G). I get this:
1. C to C (0 key)
2. G to G (0 key)
3. E to F (1 key)
4. B to C (1 key)
4. G to F (2 keys)
5. G to A (2 keys)
6. B to A (2 keys)
I would consider Cmaj7 moving to Fmaj add9 as something that's transits/flows well, because not only do you have a very small shift in keys (e.g. B to C and E to F), the second chord also retains components of its predecessor (e.g. notes C and G).
That explains the feeling some of us experience (well, music invokes different reactions) when we encounter the following chord sequence: C, Cmaj7, C7.
This is because each subsequent chord carries with it the essence of the previous chord, making the transition smooth-to-seamless, and in the case of C to Cmaj7 to C7, sounds like it's sliding smoothly downwards.
C = C, E, G.
Cmaj7 = B, E. G.
C7 = Bb, E, G.
You can get the above sequence in popular songs like Can't Take My Eyes Off You or Kiss Me.
So, for your orthodox pop song or old school rock and roll song, you can take a more direct approach by transitioning from C to C7 before you (very obviously) move to the fourth chord F major.
But if you're going indie rock or even jazz, Cmaj7 can be used in place of C. The major7 bunches up the notes in a chord, going against the grain when it comes to orthodox major and minor chord formation (i.e. your 1, 3, 5s), in which the notes in those latter chords have a somewhat comfortable separation from one another. The major7 compresses the notes, and serves to make the chord sound more like a "transition" chord, i.e. doesn't sound like the root chord, or sound quite like the equilibrium on which the entire song should be resting.
For example, in a pop song in C major key signature, we would know (and gladly take for granted) that all roads point back to C major because it is the root chord and equilibrium with which all chord sequences end. But when you use C Maj7 to open or close a line of chords or a movement in the song (e,g. verse, chorus), it doesn't immediately assure the listener that it is the main chord of the song and, I believe, would impress upon the listener that something is definitely coming up next, hence my labeling it as a transition chord.
Same case when moving from G back to C in C major key signature. Some use G7 (G, B, D, F), and others use G and transition it to G aug5 (G, B, D#). The destination is the same, i.e. C major (C, E, G), but it is all about finding the variation of the G chord that would make it a lot closer to its following chord.
In trying to return to C major chord, how am I going to make that journey back to the E note (which is part of C major chord)?
The closest the G major chord has to offer is D, which is 2 keys removed from E.
G7 has F, which is 1 key removed from E. So you could "fall back down" to C major from G7. Alternatively, G aug5 has D#, which is always 1 key removed from E. This allows you to "crawl back up" to C major from G aug5.
I don't know what's the accurate term that best captures the "jazzing up", "fuzzying" of chords, but depending on the effect you want to get out of the song, certain chord variations will help. This is especially useful for an amateur musician like me, who has in the earlier stages of songwriting, made chord sequences out of orthodox 1-3-5 chords, before adding variations.
Let's use "Count of Me Singapore" as an example. It's in the key of F. If you're an old school pop songwriter who loves clean and orthodox chords, this would be your chordwork.
You and (Fmaj) me
We'll do our (C) part
Stand to- (Gm) -gether (C)
Heart to (F) heart
We're going to (Bb) show the world
What (Bb/C) Singapore can (F) be
We can achi- (Bb) -eve (C)
We can achi- (F) -eve
The arrangement mostly relies of first (F), fourth (Bb) and fifth chords (C), comprising the clean and orthodox 135.
Now, let's do an extreme make-over with jazzed up chords to achieve a dreamier effect on the song, which it doesn't really need because it's already a very well-written song. That "Bb on C" chord in the original song alone, in my opinion, the genius stroke that at the same time holds together and defines the song.
(C7 sus4) You and (Fmaj7 add11) me (Fmaj7)
We'll do our (C7 sus4) part (C7)
Stand to- (Gm7 add9) -gether (C7)
Heart to (Fmaj7) heart
(F7) We're going to (Bb) show the world
What (Bb/C) Singapore can (Am7) be (Dm add9)
We can achi- (Gm7) -eve (C7 sus4) (C7)
We can achi- (F) -eve (F6) (F7)
For the heck of it, I'll put in the chorus:
(Bb) Count on (Bb/C) me Singa- (Am7) -pore (Dm add9)
(Gm7) Count on (C7 sus4) me Singa- (Fmaj7) -pore (F7)
(Bb) Count on (Bb/C) me to give my (Am7) best and (Dm add9) more
(Gm7) Count on (C7 sus4) me Singa- (F) -pore