What is the difference between a BA9s base, BAY9s base and BAX9s base?

Bayonet based abbreviation is BA followed by a number, which represent the base diameter in mm.

BA9 base is a Bayonet base with a diameter of 9mm (0.35in)

BA9s (single contact)

The BA9s base has a diameter of 9 mm with one contacts on the bottom and two parallel lugs on opposing sides of the base 180 degrees apart.
BA9s
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BAY9s bayonet base

When the two parallel lugs are at 120 degree apart the base is named BAY9s
BAY9s

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BAX9s bayonet base

When the two parallel lugs are at 150 degree apart the base is named BAX9s
BAY9s

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