musiclover,
i've always understood the identity in just the way you stated; God's essence is identical with His existence. for 'existence itself', as Avicenna would have said, is ambiguous between (1) existence with the condition of no addition and (2) existence without the condition of this addition. (1) is God's existence; for, being perfect, God's existence positively excludes any addition. (2) is being qua being (wujud 'aama/esse commune, in the Arabic and Latin respectively), which doesn't include any addition but can do so. And importantly, (1) =\= (2).