We saw energy/lightning being produced uncontrollably when Kylo bled his kyber as well: https://i.imgur.com/E8LlMLK.jpg
So yeah, it definitely happens as a byproduct of the bleeding process itself.
As for the environmental lightning storms: keep in mind that Vader was also bleeding his crystal on the dark side locus/nexus on Mustafar, which allowed him to reach deeper into the Force than ever before: https://i.imgur.com/fMeSAEe.jpg
So even IF Vader was inadvertently causing those storms, it's certainly not something he can do without being significantly amped.