I don't understand this about Strings and equality.

The double == operator checks primitives for equal values, however with Objects it compares the memory address of the operands instead. Strings are objects and that is what the diagrams you included are highlighting. Strings are objects with a bit of a curve ball in that the String pool was implemented to save memory by pointing duplicate Strings to the same memory address. This can be over ridden by just specifying that you want any same String to have a unique memory address by using the "new" keyword and creating like any other Object. If we implemented a String the normal way and then creating a new String(), for a unique memory address, with the same text to compare using the equality operator the result would be false as the two Strings have different memory addresses:

// two strings with "test" as their value
String one = "test";
String two = new String("test");

System.out.println(one == two); // false

Also the methods of the String class return "new" Strings (the methods that return Strings, obviously the .length() method does not). This is why you would get false if you compared the results of the same method being called twice on one String as in the example of .toUpperCase() from your question. Another example showing the same thing with a different method call:

String one = "test";
 System.out.println(one.substring(1) == one.substring(1)); // false

This is the important part to remember ↓ If you want to compare String values, or compare any Object, always use the .equals() method. Otherwise you are comparing memory addresses and this can have unexpected results if you use the String classes methods.

// two strings with "test" as their value
String one = "test";
String two = new String("test");

System.out.println(one.equals(two)); // true

String one = "test";
String two = new String("test");

two = two.intern()
System.out.println(one == two); // true