I think there are (probably) TWO candidates
Buatier de Kolta (1847-1903)
Pr. Erwin (1861-1946)
I don't know WHEN Pr. Erwin began to perform but in France we know that Buatier began in 1870 and became a celebrity in England in 1875.
His first contract was 1st may 1875 at Egyptian Hall and immediatly, it was a success (see Bell's Weekly Messenger)
He did the silk vanish like this :
Excerpt from L'Illusionniste N 22, oct. 1903, page 181 :
Disparition de Foulard
Systme Buatier de Kolta
On possde un bout de doigt creux de deux ou trois centimtres de long, pouvant s'adapter, comme un d, l'extrmit de l'index, et peint de faon tromper l'oeil du spectateur. Un foulard ordinaire est prsent, roul entre les mains, les manches tant releves, et dpos en apparence, dans la main gauche, mais il est laiss dans la droite et abandonn sur la servante en venant chercher la baguette. La main gauche est entrouverte et on voit encore un peu de foulard. C'est un petit carr d'toffe semblable, de dix centimtres de ct, qui tait d'abord cach dans les plis du vritable foulard et qu'on a gard pour produire cette feinte. Aprs l'avoir introduit dans le faux doigt, on met celui-ci au bout d'un de ceux de la main droite et on ouvre lentement la gauche qui est cense avoir conserv le foulard.
That's to say :
A Silk Vanish
Buatier de Kolta's way
One has an end of hollow finger of two or three centimetres length, being able to adapt, like a die, at the end of the index, and painted in order to mislead the eye of the witness. An ordinary silk is presented, rolled between the hands, the handles being raised, and deposited seemingly, in the left hand, but it is left in the right hand and is load on the servant while coming to seek the magic wand. The left hand is half-opened and one still sees a piece of the silk. It is a small similar fabric square, of ten side centimetres, which were initially hidden in the folds of the true silk and which one kept to produce this pretence. After having introduced it into the false finger, one puts this one at the end of one of those fingers of the right hand and one slowly opens the left which is supposed to have preserved the silk.
In the Magic Wand, Dec 1934, we can read this about Pr. Erwin :
THE ORIGINAL THUMB TIP
A recent visit from Prof. Herwin, of Bristol, recalls the fact that he is the inventor of the useful gadget the thumb-tip. Some time prior to 1885, Prof. Herwin, attaching a scrap of silk to the inside of a false finger, devised the well-known proof that a silk in the hand is still there when the mouchoir has really disappeared. Then he thought of the thumb tip and it was one of the earliest models, if not the earliest, which was shown to us on its way to the museum of the Magic Circle, where that fake, which has helped to make many a magical reputation, now reposes."
For the idea of using a false finger, it seems that Buatier precede Erwin, therefore he can have also invent(?) (Adapt ?) the thumb tip.