Re-reading the famous Carry article by Koijen, Moskowitz, Pedersen, Vugt that you linked (at least the online version) I find that they define Carry of foreign currency as (equation 7 on Page 9):
$$C_t = \frac{S_t-F_t}{F_t}$$
However, I have also seen sources where the carry is defined (like you did) as
$$C_t = \frac{S_t-F_t}{S_t}$$
In both cases Carry is intended to describe the return on forward long position of the foreign currency if "all things remain the same", i.e. in this case if the spot rate remains the same (i.e. $S_T=S_t$, where t is now and T is the maturity date of the forward). The difference between these two definitions is in the denominator, i.e in the "amount invested". With a derivative like a forward, the "amount invested" is unclear to begin with; Pedersen assumes that you set aside the full value of the forward contract, while the other definition assumes you set aside the amount S, the spot value of the foreign currency. Either convention will do, and S or F are of the same order of magnitude in any case.
Now we come to hedging foreign bonds with forwards. Since you are long bonds, you are long the foreign currency, to hedge your FX risk you must short the forward, i.e. sell the foreign currency in the forward market. The amount of foreign currency you sell should be your forecast of what the bonds will be worth at time T. Usually this will be a bit higher than the current value of the bonds, since the bonds will accrue some interest, which we can assume will be kept in the foreign currency and/or re-invested in foreign bonds. If you are wrong in your estimate you will have a "hedging error" (an undesired impact from foreign exchange fluctuations) but this will be quite small in most cases.
People refer to $h_t$ as the "hedging cost" but this is confusing terminology. For one thing $h_t$ can be positive or negative, unlike the everyday meaning of "cost" which is always a detraction from your wealth. The hedge leg will earn a return over time (the "hedge return"), which depends on the movements of the spot rate. $h_t$ is the hedge return when the spot rate remains unchanged ($S_T=S_t$). With this definition we see that this is exactly the same as the carry, but of the opposite sign, because the hedge is short the foreign currency, while the carry is calculated for a person long the currency. So $h_t=-C_t$.
Why do market participants "often take the interest rate differential as the hedging cost"? It is an approximation. We have $h=(S−F)/S=1−F/S=1−\frac{1+r_e}{1+r_u}=\frac{r_u−r_e}{1+r_u}\approx r_u−r_e$ because the denominator $1+r_u$ is close to 1. It is easy to subtract two numbers, and although wrong, it is usually not too far off.
Q1. What exactly do you mean with "$h_t$ is the hedge return when the spot rate remains unchanged" ?
When you insure an automobile for 1 year, the "cost of insurance" refers to the fact that you have to pay H to the insurance company even if your car is not damaged. If you had no insurance and the car was not damaged, you would have had to pay nothing for insurance and nothing for repairs. So the "cost of insurance" is a kind of "regret" you feel when you insure fearing something bad, but nothing happens. Of course in other cases the insurance may pay off, for example if your car is heavily damaged in an accident.
By (rough) analogy, when you hedge your bonds you enter into a currency forward which will earn a return (make money or lose money). This return depends on the evolution of the spot rate and is equal to $-\frac{S_T-F_t}{S_t}$ (the minus sign is because of a short position). Suppose the spot rate remains unchanged $S_T=S_t$ then (1) You regret hedging, it was completely unnecessary (2) You still "pay a cost" (which could be a negative, i.e. a profit to you). This amount $h_t = \frac{F_t-S_t}{S_t}$ is called by some people the "cost of hedging" by analogy with the car insurance case I mentioned earlier.
Q2. they say: "For hedging you will receive domestic interest rate and essential pay foreign interest rate." Why is this the case?
Normally a company enters into a forward with a Bank. But they could also "make their own (short) forward" by: (1) borrowing the foreign currency from a foreign bank, (2) converting the currency immediately into the domestic currency at the rate $S_t$, (3) Investing the domestic currency at the domestic rate until time T. In fact the proof of the CIP Theorem makes use of the fact that the Bank forward and the "do it yourself" forward are equivalent and must be priced the same. Because the foreign and domestic interest rates are not equal this will usually result in some gain or loss (except in the case where the future spot rate turns out to be exactly equal to the forward rate F i.e. $S_T=F_t$, when it will be zero).
[To help remember this, you might imagine a very troubled emerging market country with very high inflation and very high interest rates. In these cases people say "Oh my god, it is so expensive to hedge country X". What they mean is that $r_{foreign} >> r_{domestic}$ and $F<<S$ so hedging amounts to borrowing expensively and investing cheaply, or selling the currency forward at a poor value (i.e. below the current value). And it is true that if (by some miracle) the exchange rate does not devalue you will pay a big cost of hedging. But of course the market is not stupid and in these cases there is a high chance of devaluation, it is practically a certainty sooner or later.]
