Howdy,
You will be pleased to know that what you have is anything but a "weed of the rose world", but rather an Antique Rose (aka: Old Garden Rose) of value and great merit.

This is indeed a Moss variety, as you have skillfully noted. Unfortunately, what is less easy to determine is precisely which variety it is. A most important identifying feature is one that you are unsure of: whether it reblooms later in the season or not. That one feature alone will narrow down the possibilities. If it does not rebloom later in the year, then the possibilities are numerous. There are dozens of double pink Mosses, many of which are often collectively (and not always correctly) called "Old Pink Moss", or R. centifolia muscosa. What you have might be that rose, but it could easily be others. Judging by its shrub habit, I am expecting that you will find it is not a repeat bloomer. However, if it does rebloom, there are three most common varieties of reblooming Mosses. (Also known as "Mossy Remontants") They are: 'Rene d'Anjou', 'Salet', and 'Soupert et Notting'. All are double pinks with some late season rebloom. But as I say, judging by your specimen's growth habit, I expect this is one of the non-repeating OGR Mosses.

In either case, you have a worthwhile rose of historical value. Most of the Mosses were bred in the mid to late 1800's in Europe (ostensibly France) and were Victorian favorites. The ones we still find in our modern gardens are born survivors and likely the best of the lot. Most of them have very good disease resistance and most are Winter hardy and survive on relative neglect. Sounds like good qualities for a garden shrub, yes?

I hope that sheds some light on your rose. See the link for more info on the Mosses.

Regards,
Paul

Here is a link that might be useful: More about Mosses

Thank you so much for the information and the link. I am so excited to have a jumping off place to learn about these roses! I will be keeping a close eye on how and when they bloom this year and reading when I get the chance. I see there's a forum here for antique roses, so that should be a good place to check in, also.
Again, thanks for taking the time to point me in the right direction.