Can anyone explain to me what the difference is between a hash chain, transaction chain and a blockchain?
Difference = none.
A block chain is a linear series of data inter connected by cryptographic relationships. In your case it would be sha256.
What constitutes data is entirely irrelevant and totally data agnostic. Do not confuse a generic block chain with Bitcoin. The data does not have to be transactional or singular. It could well be photos of my pet tortoise eating goldfish.
The block is just a thing. The data within could also be individual SQL transactions, multiple transactions, or if you have a proper database the recovery logs. You can shove whatever you want into the chain. This is a cutting edge of IT research/marketing.
The important thing is that the chain 's integrity can be maintained and verified by cryptographic hashing, with all of the usual security proofs.