This has been a complex topic for so many people, so it’s important to simplify it here—separately from family vocabulary: simple to complex names to call your relatives. In all ramifications, these concepts are different and are in no way synonymous.
Relax, you will definitely understand them perfectly with this simple analysis:
If your mum remarries and gives birth to a child, the child is your half-brother or half-sister because the person who gave birth to you gives birth to him or her. Likewise, if your father remarries, the child he has through the new wife is your half-brother or half-sister because the child is your father’s.
However, the child your mother or father’s new partner brings from a previous marriage is your step-brother or step-sister because it’s the union that brings both of you together, not that any of your parents gave birth to them.
Also read: Family vocabulary
In polygamy, the child of your father’s other wife is your half-brother or half-sister because you share the blood of one of your parents (your father’s). Though this aspect of English language isn’t based on the concept of polygamy, this explanation is correct and accurate.
Basically, your half-brother/half-sister shares 50% of your parents with you, but your stepbrother/stepsister ‘stepped’ into your life from somewhere else.
Succinctly, you are related to your half-brother/half-sister by blood but to your stepbrother/stepsister through marriage.