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I get the difference between 1/8'' and 3/32'', but does this 1/2'' really make a difference, or is it the same thing?

I want to use the chain for track cycling.

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    Just use cogs and chain with matching widths. You can get away with using a narrower cog on a wider chain if you had to in a pinch but it's not ideal. You can't use narrow chain on wider cog though. Being old school all my track cogs, rings and chain are 1/8". As to there being a difference, I suspect in the early days the wider chain and cogs were considered stronger and more suitable for sprinters but now it's about transmission efficiency and aerodynamics so I guess drivetrain components are made with those factors in mind. BTW the difference between them is 1/32", not 1/2". Nov 29 '19 at 22:03
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That 1/2" is a pitch, ie. distance between 2 pins, which is standard on all bikes, expect, maybe, some super vintage ones. Just forget about it and pay attention to width (1/8 or 3/32").

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The measurement is the distance between inner plates of the chain. If your cogs are thicker than 3/32", you need 1/8" chain. Traditionally track bikes have used wide chain, but since the 2000s there have been cogs at both widths.

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