The use of Roman Numerals started after the Baroque so what is written in the original question is a more modern usage (it's pretty good in itself.)
In the Baroque, the bass note (not the chord root) is indicated on a staff. Then intervals above that bass are indicated with numbers following a few rules. The third is not indicated unless it has an accidental. The fifth is omitted in root position. The key signature is important.
In the key of C major, the root position chords are just the notes. In the first inversion, the bass is indicated so a note E with the symbol 6 means E, its third G, and its 6th C and is thus the first inversion of a C chord. Most of the theory in the Baroque was written in terms of intervals. The second inversion of a C major chord would be a G with 64 over it. (Modern use is to indicate the chord root by letter and the chord form from the figured bass, also quick to read but not the same thing.)
A sharpened interval is sometimes written with a / through the number or a + or a # before the number. A flattened interval is written mostly with a b before the number (flat sign), rarely with . A # or b with no number applies to the third above the root.
A C augmented chord would have a C note ($100) in the bass and a #5 as the figure, C#5. In C major, a C minor chord would be C b (I really have to figure out how to insert staff notaton.) C diminished would be Cb5b. Note that B diminished is just D6 (D, its third F, and its sixth, B). It's close to being a keyboard tablature.
This site has an introduction. http://openmusictheory.com/thoroughbassFigures.html
There are other books, Arnold.s and CPE Bach's come to mind.
Quick list of augmented chord: C augmented C not #3, first inversion E# note 6
, second inversion G# note, 64. Italian Sixth in C major: Ab on the bass line 4#6.
