I had same problem with magento rest api, looks they have problem in core (And they didn't fix this problem yet).. But I found solution with filters, you can filter by website_id (Please set website_id in filter no website_ids) and rest api will filter result by website_id).

Example code is here
In this case I set website_id = 2

```
    https://your-site/rest/all/V1/products/?
    searchCriteria[filter_groups][0][filters][0][field]=website_id&
    searchCriteria[filter_groups][0][filters][0][value]=2&
    searchCriteria[filter_groups][0][filters][0][condition_type]=eq
```