I honestly don't think you will find many people trying to give an earnest, precise, definition of "forward momentum" in music. It's quite a general and subjective term - a bit like saying that something is 'upbeat', or 'laid back', or 'soulful'.
Nevertheless, perhaps we can have a look at some references to forward momentum in a variety of sources, and see if we can find any commonality in what they are referring to:
Tension and release is a basic fundamental of composing music. Even a melody as simple as do-re-do captures the essence of tension and release. You start at home, you go somewhere, you come back.
A common problem I hear in many of my students' work (and one I am guilty of as well), is having too much resolution and not enough tension. I don’t mean too many consonances vs. dissonances, I mean for the overall forward motion and flow of the music there are too many moments where everything feels finished.
In rhythmic or formal terms, short-term repetition builds up tension, which is released with the change in pattern. The forward momentum of the sequence comes from this constraint and break-out pattern...
An Alternative Temporal Approach to Jazz Improvisation in the Music of Andrew Hill:
The emotional effects of polyrhythm may vary (including uneasiness, humor, freedom), but this rhythmic device invariably creates a general perception of tension, an anticipation of resolution, and a sensation of forward momentum.’17
Dissonant harmonies can make listeners feel unsettled and tense. This tension makes listeners wish for the release and resolution of consonant harmonies. When dissonant harmonies change to consonant ones, it is called resolution. Longing for resolution can give music a sense of forward momentum, and the way that resolutions are granted or withheld can make listeners feel either satisfied or frustrated.
as you play through your verse, you might choose to play louder as you go. That generates forward motion (momentum), because listeners subconsciously want to know where that crescendo of sound will lead.
Psychology of Music, ed. by Diana Deutsch :
Within Western tonal music, incidentally, the hierarchical organization of the levels of musical interpretation noted in the preceding section—in addition to having different associated time constants—contributes different and sometimes recursive levels of dynamism to the music. Thus, even after a melody has itself come to a stable resting point, the harmony may maintain tension by delaying resolution to the tonic; and even after the harmony and the melody have both subsequently resolved to the locally defined tonic, an implicit tension may remain at a deeper level until that locally established tonic finally modulates back to the initially instated global tonic of the piece as a whole. The forward momentum of music (as of life, generally) is thus maintained by a hierarchy of subgoals within subgoals
Dubstep Drumming By Donny Gruendler :
Listen to any Dubstep track. What makes the song groove and flow as you are listening to it? Does is feel like a train that cannot be stopped? Is the rhythm so incessant and repetitive that it could groove for days without wavering in its consistency? Are you tapping your foot without even realizing it? If you answered "yes" to any of these questions, then you also realize the exact reason why many dub artists program syncopated hi-hat patterns alongside a static half-time drum groove: They add a relentless, repetitive, and unwavering forward momentum to a conventional kick and snare pattern. As a result, you must master a few genre-specific sixteenth-note hi-hat partials, which will add momentum and an inherent drive to your grooves.
I think there is some commonality in the ideas there - maybe enough to attempt a definition: forward momentum is a sensation that comes from the (possibly repeated) building of tension/anticipation, followed by the release or resolution of that tension. It seems that this can happen in a number of ways - through harmonic tensions and releases; through repetitions that are expected to continue (as each successive repetition of a rhythmic figure builds the expectation that it will repeat again); repetitions that aren't expected to continue (short-term repetition builds up tension, which is released with the change in pattern); or increases in volume.
You can probably think of other devices that build anticipation - the addition of additional instruments into the mix building up to a significant climax; sounds with slow attacks that build up to a climax, such as reverse cymbals, or vocal rises / reverse reverb effects; drum fills that build anticipation of the start of the next bar or section; even a small ritardando that creates a slightly longer moment of anticipation before a climax - all of these could be said to contribute to forward momentum.
On the subject of swing, you found two quotes that seemed somewhat contradictory - one saying that evenly-spaced beats provided forward momentum, while another saying that forward momentum was provided by swung beats. On actually listening to the example, Tommy is talking about playing nice even quarter notes (and he goes on to swing his eighths). On the other hand, some other quotes I found suggested that straight eighths (on instruments other than drums) can provide forward momentum - from Why Do Jazz Musicians Swing Their Eighth Notes?:
The common use of relatively "straight" eighth notes by improvising soloists helps to sustain forward momentum, whereas the less even, triplet-like "swing" eighth notes used more frequently by drummers facilitate the perception of a quarter-note beat
while Jazz Theory and Practice By Jeffrey Hellmer, Richard Lawn is another source associating the same concept with swung eighths:
The triplet, or 12/8 feel... provides a visual reinforcement of the same form of push, or forward momentum (anticipation), in the middle and end of each measure
Hmm. it seems that the relationship between 'swing' and forward momentum is a bit harder to pin down. Perhaps an example will help:
The verse section there is played straight, while the chorus is swung. Which has more 'forward momentum' and anticipation? I think it's hard to say - they both have a drive to them, just in a different feel, in my opinion. The phrase "forward momentum" does come up a lot in writing about jazz, and I wonder if there's some confusion of causation and correlation going on; Jazz certainly often has forward momentum, and it also often has swing, but I didn't find anything to suggest convincingly that forward momentum is the preserve of swung rhythms (or percussive pieces).