TLDR:
The jump frequency depends on how you specify the jump size distribution. If you want the $\lambda$ to actually represent the jump frequency under a certain jump-diffusion model, then you should jointly estimate all model parameters, e.g. using maximum likelihood estimation (MLE) or generalized method of moments (GMM).
Example:
Consider a general jump-diffusion model for the logarithmic asset price process $X_t = \ln \left( S_t / S_0 \right)$
$$
X_t = \gamma t + \sigma W_t + \sum_{i = 1}^n Y_i
$$
where $Y_i$ are the i.i.d. jump sizes and the Poisson process $N$ has a intensity $\lambda$. You can relatively easily compute the log-return density via e.g. the Fang and Oosterlee (2008) COS method and then run a MLE jointly for all model parameters. Consider e.g. the following specification of the density of $Y_i$:
$$
f_Y(x) = p \eta_+ e^{-\eta_+ \left( x - \kappa_+ \right)} \mathrm{1} \left\{ x \geq \kappa_+ \right\} + (1 - p) \eta_- e^{\eta_- \left( x - \kappa_- \right)} \mathrm{1} \left\{ x \leq \kappa_- \right\}.
$$
original model: For $\kappa_+ = \kappa_- = 0$, this is the Kou (2002) double exponential jump diffusion model. The jump size density has two exponential tails that start at the origin.
displaced model: For $\kappa_+ > 0$ and $\kappa_- < 0$, the two tails are being displaced away from the origin.
If you estimate both of the above models using MLE then you will find that under the displaced model:
The diffusion coefficient $\sigma$ is larger and
the intensity $\lambda$ smaller.
The reasons for this is that in the original model, both the diffusion and the jumps generate small return noise. Thus you need more jumps overall in the original model to obtain the same overall number of large jumps. This is compensated for by a smaller diffusion coefficient.
The below plot illustrates this using the Levy measure. It was generated by (i) fixing the parameters of the displaced model,
(ii) simulating a time-series of logarithmic returns and then (iii) using MLE to infer the matching original model parameters. We see that the tail behaviour is almost identical. The original model also generates jumps in $\left[ \kappa_-, \kappa_+ \right]$ and thus needs a higher frequency of them.
