The verb "to like" is perfectly happy with either gerund or infinitive objects, with no real difference in grammatical or semantic relationships. Not every verb is quite so flexible.
The first example on the page that you have cited is "She enjoys being photographed." Even though "to like" and "to enjoy" have similar meanings, "to enjoy" is not happy with infinitive objects. To the native ear, the sentence "She enjoys to be photographed" sounds foreign, awkward, or just plain wrong.
Another verb that is less flexible than "to like" is "to want". In this case, infinitives are good objects. "I want to be understood" sounds perfectly natural, but "I want being understood" sounds bad.
These objects themselves, the infinitives and gerunds both, have nothing to do with expressing preference. The verbs handle that. The verbs also govern the kinds of arguments that they take.
The infinitives and gerunds in question happen to be passive. There do exist active voice versions of each. The active -ing form (which is used for both gerunds and so-called present participles) is simply that -- the -ing form. The active voice equivalent of "being photographed" is "photographing". The active infinitive form is, well, just the infinitive form. The active voice equivalent of "to be photographed" is "to photograph".
In both cases, the passive is constructed from the appropriate form of "to be" with a so-called past participle -- doing or being done, to do or to be done.