Taking a look at soft forks I understand that soft forks are initiated by miners upgrading to a stricter set of rules. Blocks generated by miners operating under the new rules are recognized by nodes running under the old set of rules.

If the a majority of miners switches to the new (stricter) rules, the blockchain will only consist of blocks generated by them at some point. This is due to the fact, that miners running under the new set of rules reject blocks mined under the old set of rules, but nodes operating under the old set accept their blocks.

If only a minority of miners switches to the new rules, there is the risk that the blockchain ultimately forks. We end up in a situation similar to a hard fork. Miners running on the new software will only recognize blocks mined by miners who have updated and reject blocks that have been mined by miners running under the old set of rules. Since miners who have updated their software are in the minority, their chain is not the longest chain. It will therefore be disregarded by miners operating under the old set of rules which results in a permanent split of the blockchain.

Is that understanding of a soft fork correct?