I hear many native speakers say, for example, "glad to connect" or "Not sure about that". Where they delete "I am" at the beginning of the sentence, Is that kind of "short the talk", I mean to speak a fewer words, or it has a conditions/rules to delete those words?
In informal speech, pronouns may sometimes be removed in sentences, together with some other words, especially copulas and auxiliaries:
[Have you] ever been there? [I'm] going to the shops. [Do you] want to come?
Seen on signs: [I am/We are] out to lunch; [I/we will be] back at 1:00 [P.M].
In speech, when pronouns are not dropped, they are more often elided than other words in an utterance.
You asked whether there is a rule or not. As you see in some examples presented above, this type of removal mostly happen to 'you' as the subject and the auxiliary preceding it in yes/no questions e.g.
'[Do you] want to come?'
and 'I' when it's the subject in a sentence and the auxiliary following it e.g your friend's calling your name and you say,
1In your examples, IMO it's not useful to put things like [P.M] or [of it] inside brackets. Those demonstrate ellipsis, but this question is about a special kind of ellipsis. Not bracketing them would demonstrate your point better. +1– M.A.R.Mar 18, 2016 at 14:21
You're right, ☺ though I wanted the writing to be more learner-friendly so readers could make a connection.– YuriMar 18, 2016 at 14:26
This answer is more easy to read, has many examples and more "learner-friendly", however The answer of @IͶΔ is more scientifically correct. As a learner I would prefer the easiest one, and for sure, the experts would prefer the other answer.– hbakMar 18, 2016 at 17:30
This is called conversational deletion.
To quote the excellent answer from JLawler, (quote from Thrasher, Randolph H. Jr. 1974. Shouldn't Ignore These Strings: A Study of Conversational Deletion, Ph.D. Dissertation, Linguistics, University of Michigan, Ann Arbor)
The phenomenon can be viewed as erosion of the beginning of sentences, deleting (some, but not all) articles, dummies, auxiliaries, possessives, conditional if, and -- most relevantly for this discussion -- subject pronouns. But it only erodes up to a point, and only in some cases.
Whatever is exposed (in sentence initial position) can be swept away. If erosion of the first element exposes another vulnerable element, this too may be eroded. The process continues until a hard (non-vulnerable) element is encountered." [p.9]
In general, exposed first-person subjects are vulnerable in statements, and second-person in questions, and any exposed pronoun is vulnerable if it is recoverable from later in the sentence.
See also Can even an expert use Conversational Deletion in his official place?
Thanks @Alan; I didn't find that. However, flagging for closure is enough, you don't need to comment on my answer.– M.A.R.Mar 18, 2016 at 14:58