Anaphylaxis (the allergic reaction that has happened after a few injections) does release many of the same inflammatory proteins as a cytokine storm, yes. The specific ingredient in these reactions has not yet been identified in any of the cases -- and may in fact be different in each situation.
As for each being a vaccine or not, the JnJ vaccine uses an older technology where a weakened cold virus is used to carry the DNA for the spike protein into cells. The protein gets made, the immune system recognizes it as an invader, and produces memory cells that fight against later exposures. The mRNA technology used for the Pfizer and Moderna vaccines consists of just the mRNA wrapped in a little bubble of fat that gets absorbed by your cells, which then make the spike protein, etc. The mRNA is much less stable than the older technology, but is still very attractive because of how well it works -- 94-95% of those that complete the two-shot course will not get CoVID-19 illness after their exposure.