Too long for a comment:

Stringham gave a talk about the content of his thesis here in the Seminar of Felix Klein in Göttingen on Monday, 1880/11/29, you can look at the scans here:

Ueber reguläre Körper im vier-dimensionalen Raum. It includes hand-drawn figures very similar to the one in the paper you linked and some that are not in the printed paper like this one:

It's written in German (presumably by Stringham's hand), and it gives the identical definition to the paper and does not talk about transitivity the way we do today.

The Felix Klein protocols do give an answer to the Question "Was Stringham at some point aware of the action of the symmetry groups?", because this talk was only the first in a series of three talks he gave in Göttingen, the other two being

Monday, 1881/02/11: Ueber diejenigen Gruppen von Bewegungen der dreifach ausgedehnten Kugel, in sich, welche die ihr einbeschriebenen regulären Körper zur Selbstdeckung bringen. [About those groups of movements of the 3d-sphere in itself, which bring a regular body inscribed in it to self-congruence.]

and

Montag, 1881/05/23: Ueber eine $\mu$-$\mu$-deutige Zuordnung einer Gruppe auf sich selbst. [About a $\mu$-$\mu$-guous assignment of a group to itself]

So while unsure if there's a bug in the original paper, I think he figured it out sometimes in 1881. It is not like he's not warning us when finishing his paper with

(The quotation is from a poem by Alfred Tennyson)

Too long for a comment:

Stringham gave a talk about the content of his thesis here in the Seminar of Felix Klein in Göttingen on Monday, 1880/11/29, you can look at the scans here:

Ueber reguläre Körper im vier-dimensionalen Raum. It includes hand-drawn figures very similar to the one in the paper you linked and some that are not in the printed paper like this one:

It's written in German (presumably by Stringham's hand), and it gives the identical definition to the paper and does not talk about transitivity the way we do today.

The Felix Klein protocols do give an answer to the Question "Was Stringham at some point aware of the action of the symmetry groups?", because this talk was only the first in a series of three talks he gave in Göttingen, the other two being

Monday, 1881/02/11: Ueber diejenigen Gruppen von Bewegungen der dreifach ausgedehnten Kugel, in sich, welche die ihr einbeschriebenen regulären Körper zur Selbstdeckung bringen. [About those groups of movements of the 3d-sphere in itself, which bring a regular body inscribed in it to self-congruence.]

and

Montag, 1881/05/23: Ueber eine $\mu$-$\mu$-deutige Zuordnung einer Gruppe auf sich selbst. [About a $\mu$-$\mu$-guous assignment of a group to itself]

So while unsure if there's a bug in the original paper, I think he figured it out sometimes in 1881. It is not like he's not warning us when finishing his paper with

(The quotation is from a poem by Alfred Tennyson)