The biological definition of Alzheimer's disease (AD) relies on multi-modal neuroimaging, yet the clinical utility of positron emission tomography (PET) is limited by cost and radiation exposure, hindering early screening at preclinical or prodromal stages. While generative models offer a promising alternative by synthesizing PET from magnetic resonance imaging (MRI), achieving subject-specific precision remains a primary challenge. Here, we introduce DIReCT$++$, a Domain-Informed ReCTified flow model for synthesizing multi-tracer PET from MRI combined with fundamental clinical information. Our approach integrates a 3D rectified flow architecture to capture complex cross-modal and cross-tracer relationships with a domain-adapted vision-language model (BiomedCLIP) that provides text-guided, personalized generation using clinical scores and imaging knowledge. Extensive evaluations on multi-center datasets demonstrate that DIReCT$++$ not only produces synthetic PET images ($^{18}$F-AV-45 and $^{18}$F-FDG) of superior fidelity and generalizability but also accurately recapitulates disease-specific patterns. Crucially, combining these synthesized PET images with MRI enables precise personalized stratification of mild cognitive impairment (MCI), advancing a scalable, data-efficient tool for the early diagnosis and prognostic prediction of AD. The source code will be released on https://github.com/ladderlab-xjtu/DIReCT-PLUS.