Demand Curve of Foreign Exchange

1. At market both price (in this case exchange rate) and quantity are endogenous ('dependent') variables.

2. You are confusing two completely distinct phenomena. Movement along demand curve and shifts in demand curve.

When exchange rate increases demand will be lower because we are moving along given demand curve - but increase in exchange rate does not cause shift in a demand curve.

In your second example when you talk about 'demand rising' you must be implicitly talking about shift in demand curve otherwise the rest of the paragraph would not make sense.

So you are actually talking about two distinct concepts. When we talk about movements along demand curve even if you reverse the relationship and 'make the exchange rate dependent variable' there will still be inverse relationship between the two. For example, if the relationship between demand for currency $(D)$ and exchange rate $(E)$ would be given by: $$D= 100 - E$$

Then 'making exchange rate the dependent variable' would mean you have to solve $E$ which would give you:

$$E=100-D$$

As you can see the relationship is still inverse.

However, you are not doing the above when you talk about 'making exchange rate dependent variable' and then asking what would happen when demand raises - you are really talking about what happens when demand shifts right which in this case would happen if in the equation $D=alpha-E$ the parameter $alpha$ which used to be in the above example $100$ would increase.

See the graph below from Mankiw's Principles of Economics that showcases the distinction between the shifts in demand and movement along demand. The graph is not about exchange rate market but you could as well just change y-axis label to exchange rate and x-axis label to quantity of currency and the point would be the same.