Interest rate delta depends on how you define it but generally (according to banking regulation anyway) is not the same as $\rho$
- $\rho$ is related to the risk free rate to do with the theoretical funding of a position assuming no arbitrage.
- Interest rate delta is related to the sensitivities present in the basket of underlyings w.r.t. different rates curves
The simplest way to understand this is the treat the PV as a function $f(\mathbf{x})$ of a vector of risk factors (e.g. rates, credit spread etc.), $\mathbf{x}$, that you represent with a Taylor expansion.
If the $i^{\text{th}}$ sensitivity is a single rate curve there exists a first order term in the expansion
$$
f(\mathbf{x}) \propto \frac{\partial f (\mathbf{x})}{\partial x_i}
$$
which you could also denote $\Delta_i$
As touched on above, what happens in risk management is that $\mathbf{x}$ is decomposed into subsets that are treated as independent (with associated correlations that can be applied later)
$$\mathbf{x} = \text{Rates} + \text{Credit} + \dots$$
and this can go further in rates as
$$\text{Rates} = 3m Libor + 6mLibor + 6mEuribor + \dots$$
A lot of risk models will currently look at a parallel shifts in the yield curve and then apply some kind of correlation factor at the various stages of decomposition above
