I'm going to slightly modify A(x) for better understanding.
A(x) = x is an apple on the table.
Statement one is correct.
Statement two does not assert - If any two objects in the universe are apples on the table ,then they must be same.
Statement two asserts that - If, every object x in the universe is an apple on the table and every object y in the universe is an apple on the table then x = y.
As a rule of thumb, remember this - There exists goes with conjunction and For Every goes with implication.