Verb + object + infinitive. What is the logic behind choosing between to-infinitive or bare infinitive?

For example, why

I saw him eat/eating pasta.

but

I asked him to eat pasta.

or

She asked him to leave her.

but

She saw him leave her.

I saw some articles about the topic that told you to remember the exact verbs. Moreover, another article — another set of verbs. Can someone explain to not native speakers what intuition lays behind this choice?

An easy rule of thumb is that the bare infinitive can only be used with:

(1) Verbs of perception:

• I saw him eat pasta.

• I heard him speak in Italian.

(2) The verbs let, make, have:

• I let them leave at once.

• I made them leave at once.

• I had them leave at once.