In the bass line you hear the 3rd of the chord which is the tonic of the song. This third is resolving (leading) in the root tone of the subdominant, which is the fourth degree of the song (IV).
Your 2nd example (notation) should be in Gb major. The natural B in the last chord is a Cb (C flat), the subdominant in Gb major.
The progression you identify is a bass line mi-fa (I6-IV), whereby the eighth note of the syncopation is the 3rd of the tonic (this note is the bass tone when we have the 1st inversion of the tonic.
In classic Roman number we write I6. This means C/E in C major, respectively Gb/Bb in Gb major or F#/A#.
(The last example has a similar rhythmic motif but the harmony is different: ii - I6).
May be you have edited your question or I have overseen this last comment to your 3rd example:
I think there might be slight differences in the chord tones between
these examples (my ear isn't the best), but they all have that
staccato chord into the longer one. It sounds so distinctly gospel; I
love this progression. I just want to understand it!
I think I have understood that what is fascinating you is rather the rhythmic figure than the chord progression itself: they all have that staccato chord into the longer one
that staccato chord into the longer one is called syncopation.
You are right: this syncopation is a specific feature of black gospel music!
The European audience was fascinated by the syncopation of the afro-american music in the second half of the last century (e.g. Mahaliah Jackson: Down by the riverside) and this is certainly a cultural feature - as well the syncopation is an element of the English language.
(I remember we sang a song that was .. .. (short long short) but it was adapted as: _.. (long short short).
Now as the other answers say it is used in passing chords, that's correct!
(The first thing I've copied from the guitar songsters was that they filled the cadence I-IV-V-I with passing tones. May be you know: This old house (Rocky Docky) or Lean on me (Someone to lean on). These tunes are both just built on passing tones between the cadence! Here you can study the passing tones and adapt them to other songs: you just play Do Do Do Re Mi Fa FA Fa Mi Fa So So and then you syncopate the bold chords and you have what you were looking for. (Mind the inversion I6 (MiSoDo=3-5-8) of the tonic!)
I can tell you other situations when this syncopation is used in Gospel music:
- F/G (this means F above G) I think that's what is really happening in your last example.
- C E7 am (E7 as dominant of the relative chord)
- C E F F#dim7 C/G (F#dim7 as secondary viio7 of G
here are some other examples of syncopations:
and here you can see how Eric Clapton sings the syncopation while Pavarotti is not so friend with it.
I always thought Elton John has been inspired by Gospel. You can discover plenty of syncopation and passing chords: