Internal variability IS unpredictable randomness as you point out.  In the *weather* prediction example, we have some ability to predict because we have information in the initial condition and know the governing equations of motion.  However, that is only helpful out to 10-14 days (tops).  In climate prediction, let's take seasonal climate (3-month average) as an example:  internal variability is those chaotic day-to-day motions that we simply cannot forecast out to ~90 days.  But we still can skillfully predict the seasonal climate b/c we have information in the boundary conditions (things like El Nino, SST anomalies, terrestrial anomalies, etc.).  Basically what is "internal" (unpredictable/random) depends on what you are predicting.  For climate prediction over a century, things like El Nino become "internal" and there are other drivers that tell us its going to warm (Greenhouse gases, etc).  

Does this help? I really appreciate you asking the question b/c this is one of those concepts I think makes sense for practitioners that have studied this for decades, but not so much for those who have not.  I think it's important to try to bridge this gap and see this from different angles.  Ask a follow up as needed!