"Definitions-Observation-Lemma-Proof" is not the right perspective for macro, and one will not, and should not, get that in a proper introduction. Start with an undergraduate text, such as Blanchard, for proper economic perspective.
(If a macro-economist wants a beginner's reference on algebraic geometry, one would recommend an undergraduate text, not Hartshorne.)
Same goes for the two sub-topics you specifically refer to---growth and international finance/international macro. Two suggestions are:
International Macroeconomics, Feenstra et al,
Introduction to Modern Economic Growth, Acemoglu.
Romer and Ljungqvist/Sargent are standard graduate texts. Romer to macro is what (baby) Rudin is for analysis. Stokey/Lucas is one of very few economics textbooks that are rigorous by mathematical standard, but it's meant to communicate some basic mathematics to economists, not for those with no economics background.