Great Soviet mathematician N.I. Akhiezer mentions in his survey article *"Чебышевское направление в теории функций"* (*"Function Theory According to Chebyshev"*) that the notation
$$T_n(x)=\frac{1}{2^{n-1}} \cos{(n\arccos x)}$$
was first introduced by S. Bernstein.

I think that the first published paper on the Chebyshev polynomials by Bernstein was *"О наилучшем приближении непрерывных функций посредством *~~полиномов~~ многочленов данной степени" which appeared in "Сообщения Харьковского математического общества", series 2, vol. 13 (**1912**), pp. 49-194. The paper is in Russian as you may guess.

In his paper, Bernstein refers to the Chebyshev polynomials as *trigonometric polynomials* which probably might explain the letter *T* in the notation.

English translation of Akhiezer's survey article is contained in *Mathematics of the 19th Century* edited by A.N. Kolmogorov.

**Edit added.** I don't know if there is an English translation of the original paper by Bernstein. This source refers to the paper as *"The optimum approximation to continuous functions by polynomials of a given power"*, Reports of the Kharkov Mathematical Society, Second Series, 1912, 13, #2-3. The original paper can be also found in volume 1 of S.N. Bernstein Collected Works (*С.Н. Бернштейн, Собрание сочинений (Том 1. Конструктивная теория функций [1905-1930])*, Москва, 1952).