What are the key differences between a liberal, liberalist, and a libertarian?

In general, when referring to contemporary United States (and to a lesser extent, Canadian) politics, (as well as often, but not always, when foreign politics are discussed by North Americans):

Liberal: refers to left of center politics, largely identified with the Democratic party. You'll often hear the term specifically associated with various issues, such as a 'social liberal', who can be regarded as being in favor of individual freedoms, such as free expression, decriminalization/acceptance of things like drugs, abortion, or homosexuality, and reduction of the influence of religion on the state. Paradoxically, the term 'fiscal liberal' often refers to strong support for the welfare state, high taxes, strong interventionist government regulation of industry and commerce, and robust legal protection of minority classes. With respect to foreign policy, liberals are often divided between pacifist/isolationist views, and strong interventionist views with an aim at the protection and expansion of human rights.

Libertarians: by contrast, are much more clearly defined, as being strongly pro-individual freedom in every aspect of governance. A libertarian would be a 'social liberal' as described above, but generally a 'fiscal conservative,' favoring greatly reduced governmental interference in commerce. Libertarians are generally distinguished from Anarchists by their strong support of a Capitalist philosophy towards economics. Extreme libertarians are often referred to as 'anarcho-capitalists' by their critics for this reason. You may also see the terms neoliberal or objectivist used as rough synonyms, though those two terms imply a stronger ideological relationship to the work of Hayek and Rand respectively.

Liberalist is not a term that is generally used. The article you link specifically defines it as a term it is using for clarity.

Note that the term liberal is also often sometimes used to refer to what many call classical liberalism, which more closely mirrors modern libertarianism. This reflects the shift in the definition of 'liberal' in the 20th century, to include a more Marxist emphasis on equality, often at the expense of classical liberalisms emphasis on freedom. In general, the European definition of liberal more closely reflects the classical definition, though over the past century, it too has shifted to a stronger emphasis on equality than it has had in the past. The definition of Libertarian, by contrast, can generally be considered to be consistent worldwide.

_{One additional notable construction is Civil Libertarian, which specifically, and exclusively refers to one who is strongly concerned with the protection of civil rights such as free speech and freedom from unreasonable searches, particularly in the political sphere. As a rule, Libertarians are by definition civil libertarians, but not necessarily vice-versa.}

Just to add a British addendum to the answer from @LessPop_MoreFizz:

For much of the 20th century, and back into at least the 19th century, one of the British political parties was named the Liberal Party. Historically, it was, at one time, one of the two major political parties.

In more recent times, it became quite small, and has variously split and and re-merged to become what is now the Liberal Democratic Party (commonly referred to as the LibDems, but sometimes loosely as the Liberals), which forms part of the current UK coalition government.

I mention this just to make it clear that the term Liberal (with a capital L) in UK politics should not be confused with the general description liberal (small l), as discussed elsewhere in this question.

I find it useful to limit the discussion in this case to nouns, and thus avoid the political connotations (and complications) of the adjectives.

liberal, n. : someone who accepts many different opinions or ways of behaving and tends to be sympathetic to other people

a. someone who believes that social and political changes should be made gradually if most people want them

Sub-definition (a) is particularly germane to the OP's question.

Merriam-Webster Unabridged lists the noun form of liberalist as simply being a variant of the noun form of liberal.

libertarian, n. : someone who believes that people should be allowed to do and think what they want

To borrow a famous software quote, liberal and libertarian are similar like car and carpet are similar.