You're confusing what happens during a soft fork and a hard fork slightly.
A soft fork is backwards compatible, meaning old nodes will still accept the softfork transactions.
Hard forks are not backwards compatible, meaning old nodes will reject the new hardfork transactions.
Think of it like this: there is a list of consensus rules. If you want to change the rules, there are two fundamental ways to do this: make the rules more strict (no longer allow something that was allowed before), or make the rules less strict (allow something new).
If you make the rules more strict, then all of the old nodes will still accept everything you are doing. This would be a soft fork.
If you make the rules less strict, then all of the old nodes will reject the new things you are doing. This would be a hard fork.