In need of some clarification on buttermilk

The author has been making their own butter and is seeking clarification on using the resulting buttermilk. They are in the UK, where cream is pasteurized, and they are not referring to store-bought buttermilk. The author buys cream, not milk from their own cow, and salts their butter as they make it. They want to know if their understanding of buttermilk is correct. They believe the buttermilk is uncultured and would need to add a culture to the cream beforehand, not after making butter. They think adding live yoghurt or shop-bought buttermilk would culture the cream, but wonder if it would need to stand at room temperature to thicken. They believe their buttermilk is essentially salty skimmed milk. They think this can be substituted for milk or water in most baking recipes, omitting added salt. They are also confused about the role of live culture in buttermilk activating baking soda, which they think might be a linguistic difference between the US and UK.