While I have not been able to find the actual date of the fireman's carabiner wrap image (descending, belaying or lowering someone using a carabiner), I have been able to find out a lot more of the surrounding information. This image shows the same carabiner, in 1880:
The carabiners originated in the Austro-Hungarian Empire (I did not find anything similar from America, where the documentation is a bit easier to find). The gourd shaped carabiners were sold by Franz Kernreuter from Vienna, Austria, first advertised for sale in 1872. Walser from Pest (now Budapest, Hungary) also made "good" carabiners. They are described on page 48 (in German) and depicted on page 262 (unnumbered):
The carabiners could hold 100 kg without bending, and were the recommended model in 1877. They are described as having an internal spring (they are one of the first carabiners to feature a spring hidden inside the gate), and a barb (dovetail latch) to keep them closed, and prevent them opening in the wrong direction. As well as being used on the belt, they are also used with a 9 mm hemp rope, for hauling up a hose, for lowering furniture or people, or for self rescue.
These very strong (for that era) carabiners were first advertised for sale in 1972:
The original purpose of the big carabiners was to pass a fireman's hose through it, so it would not be dropped, and also to have a tether from the fireman's belt to a carabiner which could be clipped to the fireman's ladder's rungs, so that both hands could be free to hold a hose. Page 16.5:
This is described on page 11 as a relatively new thing, saying that americans use a slightly different approach that does not allow self rescue (so the Austrians are using carabiners for self rescue).
So self rescue (of a firefighter) involves a rope with a carabiner (to clip up above a window or to something solid inside a room?), and a carabiner on their belt. And we already have that picture from some time after 1877, when they switched to that other type of carabiner, showing how to use it with a carabiner wrap. I have not found the method described any earlier than 1877, but in 1877 it was described as normal, while in 1869, French manuals (which go back much further than the Austrian ones) did not describe it at all.
In 1876, a spiral descender was developed for fire brigades (and subsequently sold into the 1990s!) in France. This was discussed in 1877 in Austria (they are definitely talking to each other and sharing ideas, so the French would have known about carabiner wraps if they existed):
They say that while it is a new design, they already have a "pulley" descender (they mean a horizontal spool) and it is similar to lowering someone with a carabiner and rope. The spool descender is depicted in France in 1869:
The spool is clipped above a window using a carabiner. A person gets into a bag (instead of a harness!) on the end of a rope, loops it a few times around the spool, and holds the other end of the rope, slowly paying it out so they descend. The spool stays where it is at the top. A carabiner could be used the same way, but the later picture shows they are using the carabiner wrap. That's pretty conclusive evidence.
So my conclusions:
They (France and Austria) learned how to use a spool to lower people in 1869-ish. They (Austria) learn how to use a good carabiner as a descender for self rescue, or instead of the spool for lowering, or as a belay device for lowering someone some time during the early 1870s. They get the spiral descender from France in 1877, but still prefer the carabiner wrap. The carabiners come from Austria. with the gourd ones originating in 1872. The Americans and Austrians are sharing ideas. Austrians used a tether to clip to a ladder. Americans use one to clip to the hose. Austrians can use their tether carabiner as a descender.
While I have not been able to find the actual date of the fireman's carabiner wrap image (descending, belaying or lowering someone using a carabiner), I have been able to find out a lot more of the surrounding information. This image shows the same carabiner, in 1880:
https://www.alamy.com/fire-department-fireman-with-equipment-riser-belt-rescue-rope-and-various-helmets-illustration-from-1880-feuerwehr-feuerwehrmann-mit-ausrstung-steigergurt-rettungstau-und-verschiedene-helme-illustration-aus-1880-historisch-historical-digital-improved-reproduction-of-an-original-from-the-19th-century-digitale-reproduktion-einer-originalvorlage-aus-dem-19-jahrhundert-image411744978.html
The carabiner looks like it has a torsion spring on the hinge, an approach that was invented in 1864 in America:
https://patents.google.com/patent/US45414A
It is likely to be 19 cm (7.5 inches) long, which is pretty huge for a carabiner:
https://www.google.co.uk/books/edition/Sechster_Bayr_Feuerwehrtag_in_Landshut/aigNMpibMGwC
The carabiners originated in the Austro-Hungarian Empire (I did not find anything similar from America, where the documentation is a bit easier to find). The gourd shaped carabiners were sold by Franz Kernreuter from Vienna, Austria, first advertised for sale in 1872. Walser from Pest (now Budapest, Hungary) also made "good" carabiners. They are described on page 48 (in German) and depicted on page 262 (unnumbered):
https://www.google.co.uk/books/edition/Feuerwehr_Taschenbuch/dXyrN_BVA7cC
The carabiners could hold 100 kg without bending, and were the recommended model in 1877. They are described as having an internal spring (they are one of the first carabiners to feature a spring hidden inside the gate), and a barb (dovetail latch) to keep them closed, and prevent them opening in the wrong direction. As well as being used on the belt, they are also used with a 9 mm hemp rope, for hauling up a hose, for lowering furniture or people, or for self rescue.
These very strong (for that era) carabiners were first advertised for sale in 1972:
https://anno.onb.ac.at/cgi-content/anno-plus?aid=dfw&datum=1872&page=52&size=45&qid=C5CDN8M2EJCI1P07CBLDCDUNOSK28K
The original purpose of the big carabiners was to pass a fireman's hose through it, so it would not be dropped, and also to have a tether from the fireman's belt to a carabiner which could be clipped to the fireman's ladder's rungs, so that both hands could be free to hold a hose. Page 16.5:
https://www.google.co.uk/books/edition/Österreichische_Verbands_Feuerwehr_Zeit/RTxCFhkvV3UC
This is described on page 11 as a relatively new thing, saying that americans use a slightly different approach that does not allow self rescue (so the Austrians are using carabiners for self rescue).
So self rescue (of a firefighter) involves a rope with a carabiner (to clip up above a window or to something solid inside a room?), and a carabiner on their belt. And we already have that picture from some time after 1877, when they switched to that other type of carabiner, showing how to use it with a carabiner wrap. I have not found the method described any earlier than 1877, but in 1877 it was described as normal, while in 1869, French manuals (which go back much further than the Austrian ones) did not describe it at all.
In 1876, a spiral descender was developed for fire brigades (and subsequently sold into the 1990s!) in France. This was discussed in 1877 in Austria (they are definitely talking to each other and sharing ideas, so the French would have known about carabiner wraps if they existed):
https://www.google.co.uk/books/edition/Die_Feuerwehr_Organ_für_Feuerlöschwese/vgWsL5kevOIC
They say that while it is a new design, they already have a "pulley" descender (they mean a horizontal spool) and it is similar to lowering someone with a carabiner and rope. The spool descender is depicted in France in 1869:
https://www.google.co.uk/books/edition/Moniteur_d_hygiène_et_de_salubrité_pub/CrxEAAAAcAAJ
The spool is clipped above a window using a carabiner. A person gets into a bag (instead of a harness!) on the end of a rope, loops it a few times around the spool, and holds the other end of the rope, slowly paying it out so they descend. The spool stays where it is at the top. A carabiner could be used the same way, but the later picture shows they are using the carabiner wrap. That's pretty conclusive evidence.
So my conclusions:
They (France and Austria) learned how to use a spool to lower people in 1869-ish. They (Austria) learn how to use a good carabiner as a descender for self rescue, or instead of the spool for lowering, or as a belay device for lowering someone some time during the early 1870s. They get the spiral descender from France in 1877, but still prefer the carabiner wrap. The carabiners come from Austria. with the gourd ones originating in 1872. The Americans and Austrians are sharing ideas. Austrians used a tether to clip to a ladder. Americans use one to clip to the hose. Austrians can use their tether carabiner as a descender.
Ah of course!
Wow, Deucey, you're uncovering a wealth of information! What the hell is a Pompier ladder?