You’re on the right track with your understanding of a hard fork in the context of Ethereum. Just to add to what ‘screenplay’ said, a hard fork can also be seen as a democratic process within the cryptocurrency community. It’s important because it’s like a vote where each member gets to decide which version of the blockchain they want to continue with.
Thinking of the impact a hard fork has on Ethereum, apart from creating a new currency, must also not unsee its effects on miners and software developers. Miners need to decide which blockchain they will continue to verify, and developers need to ensure that their software remains compatible with the new version. This could either divide the community or strengthen it, depending on the consensus.
To your question of whether a blockchain splits into two histories, that is correct provided both versions continue to be mined and validated. The transactions that happen post-fork on each blockchain would be unique to it, and that’s what forms the new history.
You’ve got the basic concept of a hard fork just right. It is indeed a major alteration to the protocol of a blockchain network that results in a split, creating two separate but valid versions of the blockchain’s transaction history. From the point of the fork onward, the history can differ based on which version of the rules you’re following.
When a hard fork happens with Ethereum (or any cryptocurrency for that matter), it’s because the community has decided some fundamental change is needed. Maybe it’s a security improvement, maybe introducing new functionality or even reversing transactions in extreme circumstances. These are not decisions taken lightly, because they essentially create a new currency. For example, after the DAO hack in 2016, Ethereum hard forked to create Ethereum (ETH) and Ethereum Classic (ETC).
Most importantly, for you as a user, the impact of a hard fork largely depends on how much consensus there is in the community. If everyone agrees on the changes, then the new version basically becomes Ethereum, and the old version fades away. But if there’s disagreement, like with the ETH/ETC split, then you can end up with two viable cryptocurrencies, and you need to decide which version you trust more.