macOS @syn name; begets _name or name ?

The two quotes don't contradict each other. Read the first paragraph in the section of your first quote.

By default, a readwrite property will be backed by an instance variable, which will again be synthesized automatically by the compiler.

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Most properties are synthesized automatically by the compiler. If the property is synthesized automatically, the instance variable name has an underscore at the start, _firstName in your example. If you explicitly synthesize the property using @synthesize, the instance variable name does not have an underscore at the start, firstName in your example.

I don't know why Apple chose to add the underscore for automatically synthesized properties but not for properties that are synthesized using @synthesize. If you like having the underscore for your instance variables, let the compiler synthesize your classes' properties.

When you declared a property, you can do two other things:

1. Declare an instance variable, with or without an underscore.
2. Synthesize the property, optionally specifying an instance variable.

a. If you don't use @synthesize, the property uses an instance variable with an underscore.
b. If you use just @synthesize property; the property uses an instance variable without an underscore.
c. If you use @synthesize property = instancevariable, the property uses an instance variable with the name that you gave.

If you didn't declare an instance variable matching rule a, b or c, one is created for you. If you declared an instance variable _not_ matching rule a, b or c, you have now two instance variables with different names, which will end up totally confusing you. You may get a warning.

Exception: If you implemented both the getter and setter, or if you implemented the getter for a read-only property, no instance variable will be created automatically. If you need one, you need to create it yourself.