Ragtime is a syncopated (but not 'swung') style epitomised by the first three bars of your first example, bars 2, 6, 8 of your second. The 'ragged time' occurs where notes are displaced from the quarter or 8th note grid.
Ragtime pieces DO have passages where notes are 'on the grid'. Like the main section of 'That's-a-Plenty'. Continual 'ragging' would be monotonous. But if everything was 'on the grid', it wouldn't be Ragtime!
One-step and Two-step are specific dance styles. A slower ragtime piece might be suitable for dancing a Two-step These brisk-tempo rags were considered suitable for the One-step. Remember Joplin's exhortation at the beginning of many of his Rags - 'Do not play Ragtime fast'? He might not have approved of them!
The dance most closely connected with moderate tempo ragtime might be the Cakewalk.
So yes, I dispute your definitions. The basis of all ragtime is syncopation. Played slower, you can dance a Two-step to it. Played faster you can dance a One-step.
Here's the complete 'That's-a-Plenty'
It's interesting in that, apart from the introduction and links, this version is hardly ragtime at all!
Here's another version of the same song, where although marked 'Swing' (the antithesis of Ragtime) the melody IS syncopated. Played with 'straight 8s' I'd consider it more Ragtime than the 'Rag or One-step' version you showed us!