It is clear from the quote itself that Acton is contrasting two different views of liberty. But from the larger context we can rightly describe these two views as corresponding to Acton’s conception of the Catholic view of liberty in contrast to the modern view. Thus he writes,
There is a wide divergence, an irreconcilable disagreement, between the political notions of the modern world and that which is essentially the system of the Catholic Church. It manifests itself particularly in their contradictory views of liberty, and of the functions of the civil power. The Catholic notion, defining liberty not as the power of doing what we like, but the right of being able to do what we ought, denies that general interests can supersede individual rights. It condemns, therefore, the theory of the ancient as well as of the modern state. It is founded on the divine origin and nature of authority. According to the prevailing doctrine, which derives power from the people, and deposits it ultimately in their hands, the state is omnipotent over the individual, whose only remnant of freedom is then the participation in the exercise of supreme power; while the general will is binding on him. Christian liberty is lost where this system prevails: whether in the form of the utmost diffusion of power, as in America, or of the utmost concentration of power, as in France; whether, that is to say, it is exercised by the majority, or by the delegate of the majority, — it is always a delusive freedom, founded on a servitude more or less disguised. (emphasis added)
The source of this quote is an essay on “The Roman Question” from The Rambler (January 1860), in which Acton considers the temporal power of the Roman pontiff in the context of modern revolutions.
One confirmation of the validity of Acton’s contrast, at least as regards the status of his definition of Catholic liberty, what we might identify as a basically Augustinian definition of liberty, is the appearance of this definition in an almost verbatim form in Pope John Paul II’s homily at Oriole Park at Camden Yards in Baltimore in 1995: “Every generation of Americans needs to know that freedom consists not in doing what we like, but in having the right to do what we ought.”