The reason why your Pantone 200-Coated and 200-Uncoated get a different results when you convert them to a totally different Pantone chart is simple:
The C and the U on your Pantone chart are only previews of the final printed result.
Since you selected that color on a set of printed chips, the Pantone converter simply converts that PREVIEW to another TPX that looks close to it. You get different results because the 2 previews are different!
It's not an "ink to ink" conversion because that would be almost impossible (or way too expensive) to predict the final results on all the different surfaces/material/bases these colors can be used with. The TPX chart is a totally different palette that is more "general" than the one used in printing.
Your conversion should be right and it's normal it's different.
Yes the C and U are meant to produce very similar results on different stocks but that color can be affected by other elements such as the color of the paper as well (e.g. ultra white or not.) or varnish. If you use your Pantone 200 on a dull uncoated vs Pantone 200 on a ultra bright white coated, they will look a bit different because the ink is not fully opaque and the color/brightness of the paper will influence the final result. Old paper can also be more yellowish, some uncoated dull are grayish, etc.
Unfortunately, it's not always possible to choose 2 Pantones that will look exactly the same on very different stocks. In general it's not worth using different Pantones to achieve the same result on different stocks; it's usually "good enough" to use the same Pantone on different stocks.
When someone says "Pantone 200", you look at your Pantone chart and check the coated or uncoated side depending on the stock you need to print on; coated/uncoated is for stock only, and the Pantone# is the ink. Printers won't tell you stuff like "I'll print with PMS200 U", they'll say "I'll print with PMS200" and will assume you'll look at the color book to get an idea/preview of the result.
It's up to you if you want to use C or U for your own layouts; on the plates at the print shop, it makes no difference unless you need to convert that Pantone to CMYK. The C and U are only previews for your own screen. If you need to convert to CMYK or RGB, and if you want brighter results, it's probably better to use the coated version. To my knowledge, using "uncoated" is not a standard, it's situational. I personally use coated by default even when I work on uncoated print job; I refer to my printed Pantone book for a preview, not my screen.
In general, in standard offset printing, the same ink is used to print a C or a U on coated and uncoated, unless you need something special such as a magnetic black for cheques or forms, or a fast drying ink for example. Printers can modify their recipes, use different bases/additives and often craft the Pantones themselves according to the Pantone guide.
To verify this, look at the recipe of the uncoated and coated Pantone
your printer will probably use: They are the same.
Printing on uncoated stock generally requires more ink because the stock is more absorbent. If you have a look at ink sellers, you will notice their ink estimator will give you different results for the same quantity of prints; less ink for coated, more ink for uncoated.
Printed Uncoated PMS on Coated Stock
About dot gain (absorption):