Here is a little rule: Because of the to, infinitives do not work well after most prepositions.
I think the website in question must be one of those that attempts to explain when to use an infinitive, when to use a gerund, and when we have a choice between the two. The "in order to" meaning is fairly straightforward, independent of which verb is used, but the verbs in the list use the infinitive in ways that depend what the verb might mean (e.g. as direct object of a transitive verb). ESL students may find it easier to memorize these lists of verbs than to work out the semantics involved. I really don't know myself what makes the most sense.
It is easy to imagine "do something" after each of the following:
- forget to, help someone (to), learn to, teach someone to, train someone to.
- choose to, expect to, hope to, need ((for) someone) to, offer to, want (someone) to, would like ((for) someone) to.
- agree to, encourage someone to, pretend to, promise to, recommend someone to.
- allow someone to, can/can't afford (for someone) to, decide to, manage to, mean (for someone) to, refuse to.
Note how for and to in these constructions are sometimes required, sometimes optional, and sometimes disallowed. Note also that some of these "(to) do something"s can be replaced by "doing something"s (gerunds), some with a change of meaning, and some cannot. I wish I knew how to explain the wherefores of this.