Are there two different meanings of the discount rate? Today I suddenly became aware of how the term "discount rate" are frequently used in two different contexts:


When people prefer to have something now rather than in the future. The reason is supposedly human psychology. The discount rate in this case is inherent to the individual.
The same amount of money now is more valuable than in the future. The reason is that if one has that money now, he can invest and earn interest. The discount rate in this case is the real interest rate, determined by the market.


Am I correct in thinking that these are essentially two distinct phenomena subsumed under the same term? Or there is actually a deeper connection between them?
It is standard approach by now to acknowledge the existence of a "rate of pure time preference", denoted usually by $\rho$, that characterizes individuals. This is a fundamental aspect of preferences - a "primitive" parameter. It is not a proxy for the existence of uncertainty (this is why it is found also in deterministic models), neither does it reflect "economic opportunity costs" (this is why it discounts future utility also).  

In continuous-time model this discount factor takes the form $e^{-\rho t}$, while in discrete-time models, this discount factor takes the form $\beta = 1/(1+\rho)$.  

In the individual's intertemporal maximization problem, the interest rate does not enter in the utility/objective function, but in the law of motion of personal wealth. So a negative real interest rate does not reflect on the personal discount rate.

Traditionally also, firms are considered to discount the future only with respect to the opportunity cost which in turn is assumed to be reflected in the (real) interest rate $r$. So when modelling the intertemporal problem of a firm we usually see $e^{-r t}$ and $\beta = 1/(1+ r)$.