Commercially, there were a number of vertical-cut labels in France and Britain which used inside start. "Girard & Barnes' Vertical-Cut Cylinders and Discs" names Disque Diamond and Disques Henri as Pathe subsidiary labels. The Diamond label was used in Britain, again engraved 'label' and presumably inside-start. I belive they are made by Pathe but every effort was made to disguise the fact. Also listed are Ultima, Phono_Disque and Phrynis from pre-1910, but I don't know if they're inside-start. The later generation of French vertical cut minor labels (paper-label, usually 8-inch) are edge-start. VCCD lists a number of Ideal-Aspir vertical-cuts, which it dates as 1912. Ideals (with no mention of Aspir) recorded major French artists such as the basses Belhomme and Nivette. One interesting thing is that they are not apparently pantographed (see later), and the unrecorded border at the outside does vary in width as the length of the music was different from expected. Ideal continued to make records up to the electric era, but lateral-cuts (I am assuming it's the same firm). Aspir is dated most likely 1912, though possibly back as far as 1908 - it had a copyright tax stamp on it, which came into English law in 1912, but may have been valid in Europe after the Berne convention of 1908. All recordings by the German Reichs-Rundfunk, from 1929 up to 1945, were single-sided inside-start discs.
Gray Audograph made dictation equipment that recorded and played on embossed plastic discs that were center-start and were driven by a puck which contacts the disc directly. The discs rotated on a star shaped spindle which was geared to move the discs horizontally under the stationary embossing and playback heads and the puck drive. This arrangement kept the recording surface moving at a constant linear speed under the styli and caused the discs' rotational speed to constantly and steadily decrease as the disc is recorded or played back. Although not center-start, Pemberton experimented with constant linear speed discs in the '20s. Pathe, however, seemed to be the largest commercial center-start disc producer.
Pathe discs were center-start from the company's entry into disc-producing in 1905 up until 1915. Girard & Barnes' book "Vertical-cut Cylinders and Discs" gives a history of Pathe recordings. The first discs were made in 1905 and were wax on a cement base - as the authors say, like a cylinder flattened out. The 'label' was not paper, but the details were engraved like a Berliner. The details run round the perimeter of the label area and has some similarity to the engraved label details on the edge of cylinders. The engraved lines were filled with white or ochre pigment. Obviously a paper label would not stick to a wax base and maybe couldn't be attached in the pressing process. These wax discs are phenomenally rare and in 1906 they changed to shellac, still using the engraved 'label'. Paper labels didn't come in until just about the time in 1915 when they changed to outside-start and changed the speed to a nominal 80rpm. The last center-starts had a paper label of the same design as the engraved ones but the outside-starts used a more conventional design with the Pathe 'Coq' trademark.
The reason Pathe could get their discs to end a neat half-inch from the edge is that they were all pantographed, i.e. all copied by a mechanical lever system, from cylinder. They used a master cylinder of a large size, probably the 'Concert' size, 6 inches I think. They did this right up to the end of acoustic recording. Their lateral-cut discs (such as Actuelle in Britain and the US) were also pantographed. On many of these discs, one can hear the clicking sounds of the master cylinder turning (and the rumble!) Their practice of pantographing from cylinder masters ceased in 1923.
Why start from the inside? Speed!
They are not exactly '78s' as they go from 90 to 100 rpm (officially) and in practice from 80 to 120 rpm. I've heard it said that Pathe chose to record using the center-start method because the end of a piece is often louder/higher in pitch than the start, and the outside of a disc is where there is greatest linear groove speed and the least tracing distortion and the most top. But what would happen if they recorded a short piece and didn't pantograph from a cylinder master? Pieces ending farther from the perimeter would mean that all that extra "high quality" space would be wasted, and the overall average sound quality would be worse. They could have fixed that though by timing the piece beforehand to see how far from the center to start, since many companies made many takes of each tune at each session (just incase the master stamper broke, it was said in the early days.) Now, would the public have accepted a variable, untidy, inside-start record? I suppose that since the public was used to an untidy inside on normal records that they could have accepted a variable-width outside.
There is another reason for inside start cuts, particularly on transcription disks. When you cut a disk, "chip" material that is cut away spirals up from the cut, and tends to head for the center of the disk. Professional mastering lathes vaccuum it away, but less expensive cutting systems don't have that luxury. Since it tends to go towards the center, you have less chance of it colliding with the cutting stylus if you start at the inside of the disk.
One shoe off and one shoe on
Some Radio Transcription companies produced records that were both inside and outside start. Since the fidelity and quality of the recording varried slightly from inside to outside, keeping the start the same as the ending point would preserve that quality when it was flipped over at playback time. Some companies wrote in the wax whether it was outside or inside start.
Rarity
Pathes are not rare in England, Belgium, and France, although you don't see many 78s in France in my experience. Also I've seen enough imported Pathes to believe that they are not rare in the US - scarce may be a better word. The paper label variety was also made in Canada and so do turn up.