Rich;
Where did you measure the bores to get the .702 & .706 sizes. A barrel marked 13 should accept a .710" plug gauge as 13 gauge is .710". This would be a pre 1887 proof. My Birmingham proofed W Richards (Probably J P Clabrough built) is marked with same marks except it was proofed as a 14 (has 12 gauge chamber). There is no indication it was converted from a M/Ler.
As I recall any barrel having at least .008" of choke at this point was required to be marked "Not for Ball". That .702" bore would be right on the line "IF" that is a muzzle dimension & it actually measured .710" at the breech. If that be the case it probably was originally built as a breech loader & not a conversion.
As to the two final proof marks one was as I recall applied at the time of firing & the other, known as a view mark, was applied upon inspection by the examiner. These do not imply it was proofed twice.
