Which frailty model is better for survival analysis with recurrent events and competing (teminal) risks: joint or shared?

I am going to analyse data with multiple recurring events and additional terminal event. The recurrent events are of the same kind, no hierarchy in them (like in the Prentice-Williams-Petersen). The terminal event, the "elimination from experiment" is single and precludes all over events, it is an absorbing state.

I want to analyse it with frailty models. In most sources in the Internet I noticed the use of joint two frailties: for recurrence and competing risk. But in some places, authors mention shared frailty, which theoretically means the same frailty is spanned across all subjects and both types of frailty. I don't fully understand this.

Can anyone explain, which kind of frailty model is better here?

Also, I know, that mixed-effect models have parametric assumptions about the random effects. It should be Gaussian. In frailty models I know it should be gamma. Can I check the compliance of distribution using Lilliefors adjustment for the Kolmogorov-Smirnov? The adjustment helps when I have to estimate the model parameters from the data via MLE, which would invalidate the use of the classic K-S, where the parameters must be known rather than estimated. Or I could bootstrap the K-S? Or maybe I should use the ECDF to visually compare them?

The "joint" frailty terminology, as used in the frailtypack package in R, extends the shared frailty for repeated events to include a terminal event. Separate baseline hazards and covariates are possible for the repeated versus terminal events; each frailty for the terminal event is the corresponding frailty for the repeated event but raised to a power, a further coefficient estimated in the modeling. If that power coefficient estimate isn't distinguishable from 0, then there is no evidence of frailty contributing to the terminal event (beyond the modeled covariates).