What is the logic behind using <<у меня>> in this example, rather than <<у меня есть>>?
Here's what they say in Tips & Notes for Basics 2:
Omit ”есть” if the existence of the object is obvious or not the point — very typical for describing traits (“Tom has a beautiful smile/large eyes”, “She has a very fat cat”). Also when expressing temporary states and illnesses (“She has a migraine”).
Today I asked a native Russian speaker, who got two more native speakers involved about whether it is okay to omit the есть or not, and the consensus was that it is !not! okay to omit the есть here unless this was a response to someone who already introduced the topic of water and apples. So as a general translation, this is incorrect, but as a reply to someone asking (i.e.) what you have, omitting the есть is okay because it is already known you have (or do not have) something.
"It is I" is actually the grammatically incorrect form of this statement. "It's me," is the "correct English" phrasing. You wouldn't say, "It is he," but instead you would use "him." It is the same with "me." Я is different and can actually refer to both "I" and "me," depending on the context. (Sorry, the only thing worse than linguistic prescriptivism is factually unsound linguistic prescriptivism.)