Are people being literal when they say "I like to think that …"?

I like to think introduces a pleasant or hopeful wish, but with some uncertainty.

I like to think that people are mostly good.
I like to think that I'm good in bed, but how should I know?

I like to think that is actually used literally most of the time. People like to think a certain thought, which they may actually believe (though not as a result of liking it). On the other hand, they may simply be optimistic or hopeful about a thought, yet realize that there may be evidence to the contrary. Still, the thought is nice to think.

In the following examples, the speaker likes the thought and most likely truly believes it too.

I like to think that my business plan will attract investors.

I like to think I'm a good person.

I like to think I can always count on my friends.

If the speaker adds context to the contrary of their I like to think that statement, then the speaker likes to think the thought (because it's nice and/or feels good) and generally believes it, but doesn't necessarily believe it all the time or believes that there are exceptions to the thought. This is the case in the following examples.

I like to think that I'm a good person, but I have my faults.

I like to think that I'm fit, but I could exercise more.

I like to think that I'm attractive, but I guess beauty is in the eye of the beholder.

I would like to think has a very different meaning, in that the speaker doesn't actually think the thought, but acknowledges that it would be a nice thought if he actually believed it to be true (which he doesn't).

I'd like to think that I'm good at math (but I'm really not).

I would like to think that I could take a day off and the team would be able to finish the project without me (but I don't actually think they could).

I'd like to think that all people are always well-intentioned (but too much evidence contradicts such a nice thought that I would simply be naive to believe it).

(taking the phrase as "I'd like to think X...")

I don't think that the literal meaning is "I like something so I believe it." and either way there's no matter of distrust.

The literal meaning would more likely be "I would enjoy having the thought X, but there might be evidence that contradicts the thought".

What a statement "I'd like to believe X" is intended to mean is that "I really hope X, there might be some evidence to the contrary but I'm sticking with my hope". So it is somewhat metaphorical/an understatement.