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I'm fluent in both fixed- and movable-do solfège, but I'm wondering how the "tonic sol–fa" system
I'd especially love an answer with a musical example that shows the difference between one of those two solfège systems and the "tonic sol–fa" system.
Since there are no answers and because alephzero's comment helped clarify everything for me, I'll take the liberty of writing my own. This information comes from the manual alephzero provided here. (I didn't even think to look here, so thanks!)
The tonic sol–fa system is really more a system of notation than it is a solfège system. So while the system uses solfège syllables (and abbreviations of them), tonic sol–fa functions as a notation system complete with rhythm and meter.
Tonic sol–fa is a movable do system; tonic is always "doh," the dominant scale degree is always "sol," and so on. The syllable for the leading tone is "te" (pronounced "tee"). (Most systems I see write the leading tone as "ti," the third as "mi," but tonic sol–fa has chosen the "-e" ending.) With that said, these diatonic pitches are only notated by their first initial: "doh" is "d," "sol" is "s," etc. And since it is a movable system, the key is always given at the start.
Raised pitches end in this "-e" vowel, so scale-degrees 6–♯6–7 are "l–le–t" ("lah–lee–tee"). Lowered pitches end in "-a", so 3–♭3–2 is "m–ma–r." Note that we notate the full syllable for chromatic pitches.
In contrast to solfège, tonic sol–fa specifies register by using ' or , to the right of the pitches to move them an octave higher or lower.
But the real difference is in how this system notates metered music. In tonic sol–fa, a weak beat is preceded by a :, a medium accent is preceded by |, and the downbeat is preceded by nothing (but note that barlines are occasionally inserted), so a measure of 4/4 time would be:
Key E
d :r |m :ma
If you need a little test, this notation translates to:
E F♯ G♯ G♮
More specifics about metered notation can be found in Chapter III of the manual linked at the top of this answer.
Ultimately, tonic sol–fa notation produces notation such as
The decision of whether this is a worthwhile substitute for notation, or if it is worth the time to teach, is of course up to the individual teacher.