Your maths is right, but the way you're using the terminology doesn't line up with the way people speak about this stuff in practice. Theta is used for talking about time decay on options, but the concept of time decay for options doesn't quite extend to futures and forwards (so called 'delta one' products, to distinguish them from products involving options). I've tried to explain why time decay is different for options to delta one below.
For futures and forwards, depending on the asset class of the underlying, you might hear people call dF/dt something like the 'carry'. Calling it a theta would likely cause confusion, because theta refers mainly refers to the fact that an option has smaller chance of expiring in the money when time to expiry decreases. This concept doesn't exist for futures and forwards.
Theta is also a significant factor driving PnL for an options book - especially if you are have a lot of options close to the money you can lose a lot of money every day due to theta / time decay. This isn't true for futures and forwards, where day to day changes in value are driven mainly by changes in spot price.
What makes time decay for options different to delta one
The value of an option is higher when there is more chance for it to expire in the money. There is a better chance of this if the volatility is larger and if there is more time left to expiry, because the variance of the underlying until expiry which is priced into the option is proportional to
sigma ^ 2 * (T - t). Therefore reducing the time to expiry decreases the value of the option, hence theta tends to be negative for long call and put positions. This concept only exists for options (and products involving options), as futures and forwards don't have the same concept of expiring in the money.
However, not *all* of an option's theta is due to this 'chance of expiring in the money' effect. Theta still exists for an option even as the volatility approaches zero (see the below example of a call option). This 'zero volatility' part of the theta is similar to what you calculated as 'theta' for a futures contract. But it's not the important part of an option's theta and tends to be pretty small in most cases compared to the part of theta related to the smaller chance of expiring in the money when time to expiry decreases.
Theta for a call option as volatility approaches 0
As volatility approaches zero the value of a call option becomes
- for S > K * exp(-r(T-t)): S - K * exp(-r(T-t))
- for S < K * exp(-r(T-t)): 0
(not too tricky to check this yourself if you want to).
Hence theta becomes:
- for S > K * exp(-r(T-t)): - K * exp(-r(T-t))
- for S < K * exp(-r(T-t)): 0
The case for S > K * exp(-r(T-t)) is very similar to your formula for dF/dt. In other words, roughly speaking, if there wasn't any volatility, the option theta would line up with your dF/dt formula. But in reality there will be volatility, and this component of the call's theta won't be very important compared with the component related to the smaller chance of expiring in the money when the time to expiry decreases.
Final note
In practice theta tends to be calculated as 'value in 1 day's time, assuming inputs remain unchanged' minus 'current value'. I work in the market risk team for a money manager and if I looked on my screen at work many of the instruments would have a non-zero 'Theta' value even if they aren't options. Despite this, people don't really refer to this as the 'theta' for these instruments, because of the discussion above. I think depending on the asset class it might not make much sense, might not be important, or might be referred to using different word(s).