How is "I have come" + infinitive different from Present Perfect?

Consider the following examples:

I have noticed that a lot of people are switching to Unity.
vs.
I have come to notice that a lot of people are switching to Unity.

or:

The Saddam I have come to know
vs.
The Saddam I have known

or:

"I have come to understand" vs. "I have understood"
"I have come to decide" vs. "I have decided"

My questions are:

1. How is the meaning different in the above examples?
2. I do not think it's possible to use certain verbs like "run" after "I have come to". So what sort of verbs can be used in this context?

Solution 1:

"I have come to notice [something]"

...normally emphasises the progressive nature of the action (i.e. - it didn't happen instantaneously). Per JeffSahol's comment below, in some circumstances it may imply the action was overdue (should have happened earlier), rather than that it actually took place over an extended period.

Although it's present tense, people often use "I notice" interchangeably with "I have noticed" (in respect of something recently noticed which you're now mentioning).

Thus, for example, "I came to realise Reagan was a good president", implies you only gradually realised this. Without the verb to come, there's no such implication...

"At that moment, I realised Reagan was a good president"

"I realised Reagan was a good president after studying him for many years"

Solution 2:

Come to Vinf is a common Inchoative construction.

Come to be, in fact, is usually shortened to simply become, and the troublesome word get means both come to be and come to have. The 4 sentences below all mean the same thing.

• He came to distrust what they told him.
• He came to be distrustful of what they told him.
• He became distrustful of what they told him.
• He got distrustful of what they told him.

The difference between a simple Perfect construction (1) and a Perfect inchoative (2)

1. She has recognized the symptoms in him.
2. She has come to recognize the symptoms in him.

is that in (1) - a type 3 Stative/Resultative Perfect - it's the resultant state, his possession of symptoms, that is presently relevant.

Whereas in (2) - a type 1 Universal Perfect - because it's inchoative, it's the continuing (and therefore gradual) change of resulting state that's presently relevant.

Solution 3:

"Come to" in these contexts is not a helper verb that changes the tense or mood of the verb that follows. It means to "reach or enter a state, relation, condition, use, or position". The verb that follows, in the infinitive, specifies what that state/relation/etc. is.

So, to answer your questions:

1. When you say "I have come to X", it means that you have changed your mind, or your condition has changed, or something else is different that compels you to, or at least suggests that you, X.

2. I don't think there is any rule about what verbs could follow. "I have come to run" does not make much sense by itself, at least in the sense of "come" we are talking about right now, but there are sentences where it could work.