Hi David,
Here the API doesn’t translate exactly to the UI. The checkboxes you mention control different “kinds” of branch restrictions.
When you leave them unchecked in the UI, we will create a restriction where 'kind': 'force' and 'kind': 'delete' for “Allow rewriting branch history” and “Allow deleting this branch” respectively. If you check them in the form, we don’t create these branch restrictions.
You can see all the different kinds in the schema documentation here: https://developer.atlassian.com/bitbucket/api/2/reference/resource/repositories/{username}/{repo_slug}/branch-restrictions#post
require_tasks_to_be_completed
require_passing_builds_to_merge
force
require_all_dependencies_merged
push
require_approvals_to_merge
enforce_merge_checks
restrict_merges
reset_pullrequest_approvals_on_change
delete