How are cash prizes distributed in the event of a tie for first or second?

Let's say if a tournament has $200 prize for 1st place and $100 prize for 2nd place.  Two person tie for first with the same score.  How would a "typical" tournament give out the money?  Use tie breaks and give the person with the better tie breaks $200 and the other with worse tie breaks $100?  Or do they just call it a tie (without the use of tie breaks) and give each person an equal $150 split for the top 2?  Or do they use a blitz playoff to break the tie instead of using tie break rules?

Your second assumption is correct.   Unless specifically specified otherwise, the two top players split the 1st and 2nd place prize money ($300 in your example) and each take home $150.00.

If a trophy is involved, one of those two players would win the trophy on tie-breaks, but the cash would still be split evenly.

if 1st is $100,000 and 2nd is $40,000 Then you add them togethor and split it together. 100k plus 40k is 140k dived by 2 is 70k each. This is the prize money for the millionaire chess tournament :)

Correct. Standard practice in the case of a tie is that cash prizes are divided evenly, and titles or trophies are assigned by tie-break scores.

yes