The verb "to be" εἰμί (eimi in Greek) in almost all languages is probably the most used because it is worked so hard.  Its versatility is shown clearly in John 1:1 as follows:

In the beginning was the Word = [= existence]

 . And the Word was with [the] God [= relationship]

 . . And the Word was God [= predication, in this case, a qualitative category statement]

More generally, the use of εἰμί (eimi) can be classified into at least four uses:

 - Existence, “I am.”, ie, unpredicated (see below).
 - Identification, eg, Luke 1:19, “I am Gabriel”; John 9:9, “I am [that one]”; John 10:11, “I am the good shepherd”.
 - Relationship, eg, Acts 18:10, “I am with you”.
 - Predication, eg, Acts 22:3, “I am Jewish”. (this is what some call the stative use.)

Seven times in the Gospel of John alone Jesus uses this verb to declare His eternal existence and well as identifying with the OT "I AM".

 - John 4:26 – “Then Jesus said, ‘I am.’”  [To the Samaritan woman at the well.  There is a reasonable case for this being identification, but that is a matter of taste.]
 - John 6:20 – “But then [Jesus] said to them, ‘I am.  Fear not.’”  [To the frightened disciples in the boat.]
 - John 8:24 – “If you do not trust/believe that I am, you will die in your sins.”
 - John 8:28 – “When you will lift up the Son of Man, then you will trust/know that I am.”
 - John 8:58 – “Truly, truly, I say to you; before Abraham existed, I am.”  [The Jews then tried to stone Him for blasphemy.]  Note that this and the previous two mean that Jesus, in the space of this chapter of John 8 uses the unpredicated “I am” idea in the present (v24), future (v28) and past sense (v58).  V24 & 28 appears to be tied to believers’ salvation as well.
 - John 13:19 – “From now [on] I tell you before the occurrence, that you may believe when it occurs that, I am.”
 - John 18: 5, 6, 8 – “He said to them, ‘I am.’ …Therefore, when He told them, ‘I am’, they fell backward to the ground.”  [This occurred when the Jews tried to arrest Jesus in the garden.  It could be reasonably argued that this is a case of identification.  However, the fact that the arresting mob fell backward suggests that much more is intended here.]

[There are another seven such in the other Gospels but I do not wish to clutter this too much.]

However, εἰμί (eimi) is capable of much more.  I note that both Thayer and BDAG list numerous uses for this versatile verb, the first of which is invariably meaning "to exist".

The distinguishing characteristic of stative verbs is their use to show the status of something.  "I am in love"; or, "The car is broken".  In English, this usually amounts to a simple copula between the subject and predicate (the state).  When alone, eg, "he is", amounts to a statement of existence.