That's a good sum-up but I still say your example is break-even over three hands.
This is just mathematically incorrect. If you just the river card to come and have 20% equity, that means you expect to win it 1 out of 5 times, not 1 out of 3. I'm really not sure how else I can explain it... I'm not sure how much background you have in mathematics but I would really recommend you start thinking in terms of
expected value rather than run strategy models like this. Strategy models like this require decision trees and the use of conditional probabilities.
Nope: You probably meant "expect to fill the flush by either the Turn or the River card, 1 out of 3 times."
... I'm hoping you're not trying to be a huge grammar nit because my sentence is clear as day in the context of the paragraph and is actually not grammatically incorrect... Obviously if we flop a flush draw, we are not flopping the flush. Just in case you really didn't understand what I was saying:
"Thus, even though we expect to fill a flush 1 out of 3 times (by the river) on the flop (we are looking at the hand from the flop meaning we have our two hole cards and we see the flop but not the turn or river)"
Nope entirely: The 1 out of 3 times situation ONLY exists over the two cards (Turn card + River card). It doesn't matter whether you hit the card on the Turn or the River. It doesn't matter if you miss the Turn card. The 33% equity only applies if you think in two-card chunks. It also doesn't matter when the money goes in. If zero, or a few tons of chips went in to see the Turn card, and if zero or a few more tons of chips went in to see the River card, it doesn't matter. The fact remains: IF 3 players stayed in to see both cards then you're back to 2:1 return. Even steven!
(caveat: your one flush hit has to win the pot)
This is where your thinking is flawed. It doesn't matter whether or not your flush fills on the turn or river
from the perspective of the flop. However, this is only relevant for
money going in on the flop. However, once the turn card falls, it
does matter whether you just hit the flush or not. Let's see if I can explain conditional probability without making things too complicated...
The hand is preflop and you have two suited cards (let's say the A of spades and the 5 of spades). The probability of having a flush by the river is a little less than 4%. Does this mean that we should expect our flush to fill 4% of the time after we flop two spades? No, because the board changed and that changes our probability of hitting the flush. Because we had a 4% chance to hit a flush by the river preflop, does it mean we need 24:1 odds on the flop when we flop the flush draw? No, because the board changed. If we were playing purely for flush odds, we would need 24:1 odds on our money
preflop.
In the same way, once we flop our flush draw on the flop, we have a 35% chance of hitting the flush by the river. Does this mean we will hit the flush 35% of the time once we brick the turn? No, because the turn coming changes our probability of hitting the flush. Thus, because we had a 35% chance to hit a flush on the flop, does this still mean we need 2:1 odds on the turn after we brick the turn? No, because the board changed and the probability of hitting the flush changed.
If this doesn't give you a better understanding (and I apologize if my explanations are unclear because I suck at explaining), I'll just let someone else give it a whirl because I don't know how else to explain it though if you have any specific questions, I'm more than happy to answer them.