1. One size, many names
The IEC names a cell by its chemistry and its size, so an LR44 is an alkaline button cell 11.6 mm across. Brands use their own names, and the American ANSI standard adds a three-digit code, so the same cell is sold as an LR44, an AG13 or an A76.
2. The AG names
AG13 and AG3 are Asian generic codes. AG13 is the common 11.6 mm alkaline button, and AG3 is the smaller 7.9 mm one, which is also sold as an LR41.
3. 357 and 303 mean silver oxide
Both names belong to the silver oxide cell that is also sold as an SR44. The same size in alkaline is the LR44, and the two are not interchangeable.
4. The chemistry decides, not the name
SR means silver oxide, LR and AG mean alkaline, and CR means lithium. Two cells can be exactly the same size and still be different batteries, so read the letters before you buy.
5. The three-digit name is not a measurement
The middle digit in a three-digit code is a legacy name rather than a measurement. A 357 is about 11.6 mm across, exactly like an LR44.