In the given question, two statements about the Vernacular Press Act of 1878 are provided.

Option 1 says that the Indian press failed to protest against the Vernacular Press Act of 1878. However, this is incorrect. Many newspapers and periodicals in India actively protested against the Act, which sought to control the press in the Indian vernacular languages.

Option 2 says that the Vernacular Press Act of 1878 was introduced by Viceroy Lytton against the counsel of his own Law Member. This statement is true. Viceroy Lytton did indeed insist on passing this Act in spite of resistance from the Law Member of his Council.

Option 3 suggests that both statements are correct. Given the inaccuracies in the first statement, this option too is incorrect.

Option 4 states that neither of the given statements is correct. This is not correct as the second statement is true.

Hence, the correct answer is Option 2, indicating that only the second statement is true.