Rechargeable AA Lithium

Do they exist? Are they named differently?

yup, the’re called 14500

They basically don’t. Current rechargable lithium-ion chemestries (used in 14500) have at least 3.2v (usually 3.7v), so they are not suitable as replacement for lithium AA batteries. Many flashlights support them, bypassing the boost circuit, as the led needs about 3.5v, but most other devices will be damaged by the higher voltage.

There is a lithium-ion battery (Klenti) that uses a step-down converter to ouput constant 1.5v, but the capacity isn’t better than Nimh. If you need higher voltage than Nimh there are NiZn batteries, but they are about 1.8v when fully charged, a bit higher than lithium AA.