adapted process, translation between measurable and information? Although there are plenty of questions and answers on understanding the intuition for adapted process like this post and this post I am still unclear on how an adapted filtration 'captures the information of the process up to time $t$.'
First some notation, let


*

*$(\Omega, \mathcal{F}, P)$ be a probability space.  

*$X:T\times \Omega \to (S, \Sigma)$ be a stochastic process.  That is $X_t:= X(t,\cdot)$ is an $(\mathcal{F}, \Sigma)$-measurable function for all $t\in T.$ 

*$\{\mathcal{F}_t\}$ such that $\mathcal{F}_s\subset \mathcal{F}_t$ for $s\leq t$ be a filtration of $\mathcal{F}.$ 

Definition: We say $X_t$ is adapted to the filtration $\{\mathcal{F}_t\}$ if $X_t$ is $\mathcal{F}_t$-measurable for all $t.$ 

The definition is clear as day but I am confused about the interpretation/intuition/motivation.  A classic example is the price of a stock, $S_t$ which is adapted to the natural filtration of the Brownian motion on which it is modeled $S_t = \exp(ut + \sigma B_t).$  The interpretation on the adapted filtration condition is that we only know the current price of the stock (and its history) but we don't know the future price of the stock.  

Question: (stock version): Why does the stock being adapted filtration have anything to do with knowing its current and historical prices?

I don't see the connection mathematically. In real life we have a stock and we know its price and its history.  I can't see the connection between this and that condition that $S_t$ be $\mathcal{F}_t$-measurable.  A discrete example might help illustrate my confusion.
Discrete Example
Consider the example of flipping a coin 3 times. With $X_i$ being 1 if the $i$th flip is heads and $X_i$ is 0 if the $i$th flip is tails.  Let 
$$\Omega = \{hhh, hht, hth, htt,ttt, tth, tht, thh\}$$ be the state space of the experiment and $\mathcal{F}$ all subsets of $\Omega.$  Then we define the stochastic process $X$ by $$X:\{1,2,3\}\times \Omega \to (S,\Sigma)$$ with $S=\{0,1\}$ and $\Sigma$ all subsets of $S.$ 
The smallest possible filtration to which $X$ is adapted is the natural filtration.  We compute 
$$\begin{align}
\mathcal{F}_1 &= \sigma(\{X_1^{-1}(A)| A\in \Sigma\})\\
              &= \sigma(\{X_1^{-1}(\emptyset), X_1^{-1}(0), X_1^{-1}(1), X_1^{-1}(\{0,1\})\}\\
              &= \{\emptyset, \{ttt,tth,tht,thh\}, \{hhh,hht,hth,htt\}, \Omega\}
              \end{align}$$
In this case adaptability translates to forcing $X_1$ to constant on the sets $\{ttt,tth,tht,thh\}$ and $\{hhh,hht,hth,htt\}.$  But how does this help us?  It certainly doesn't tell us which set happened and which didn't.  I can see that $X_2$ is not $\mathcal{F}_1$-measurable, but so what?  Knowing $\mathcal{F}_1$ doesn't give us any knowledge about the result of the first toss, other that it was either a heads or a tails, which we already knew.  So now I am ready to state my more general question

Question Why does the $\sigma$-algebra $\mathcal{F}_t$  'contain information' about the process up to time $t$ and what does this mean?  How can we use it to say something concrete about what has actually happened up to time $t$ and how can we justify this mathematically?

 A: It's not that $\mathcal{F}_t$ itself "contains information about the process up to time $t$" in the way you seem to be assuming.  Rather, elements of $\mathcal{F}_t$ are allowed to depend on information about the process up to time $t$.  For instance, in your example, the set $\{ttt,tth,tht,thh\}$ has some very useful information about the first toss: it tells us that the first toss was tails!
To put it another way, $\mathcal{F}_t$ should be thought of as the set of all events which depend only on what has happened up to time $t$.  So in your example, one of the events in $\mathcal{F}_1$ is that the first coin toss was tails, represented by the set $\{ttt,tth,tht,thh\}$.  Another event in $\mathcal{F}_1$ is that the first coin toss was heads, represented by the set $\{hhh,hht,hth,htt\}$.
Now, if $X$ is a random variable, to say that $X$ is $\mathcal{F}_t$ measurable means that every event we can define in terms of $X$ is in $\mathcal{F}_t$ (more precisely, for any Borel set $A$, $\{\omega:X(\omega)\in A\}\in\mathcal{F}_t$).  That is, the event of $X$ having any particular value (or any particular range of values) is in $\mathcal{F}_t$, meaning that this event depends only on what has happened up to time $t$.  So, to say that $S_t$ is adapted to the filtration means exactly that any event regarding the value of $S_t$ only depends on what happens up to time $t$.  In other words, the current price (at some time $t$) can't depend on what happens in the future.
I feel you might find this unsatisfying, and if you do, my response would be that you are simply misunderstanding the purpose of the notion of a process adapted to a filtration.  This is just a technical definition for describing stochastic processes as a mathematical structure.  It doesn't have any special powers and isn't going to magically answer any questions about stock prices.  Treat it as just a definition and nothing more, and be patient until you see applications where the definition is useful.
