Short answer
If the device or empty cell says CR2032H, buy that grade when available. A standard CR2032 may fit but under-perform on pulse / high-drain loads.
Both are primary (non-rechargeable) lithium manganese dioxide — not LIR2032 or ML2032.
What is the same
Approximate size (20 × 3.2 mm), 3 V nominal primary CR chemistry family, disposable.
What is different
H denotes a high-drain CR2032-class variant used by some manufacturers. Availability and naming vary by brand.
Capacity figures overlap and are manufacturer-dependent — do not treat H as a different thickness or voltage class.
Do not use list
Do not substitute LIR2032 or ML2032 for CR2032H (or CR2032) by size alone.
Do not flatten CR2032H into a pack label for BR2032.