According to the General Data Protection Regulation (GDPR), a data processor is a natural or legal person, agency, public authority, or any other body who processes personal data on behalf of a data controller. A data controller is a natural or legal person, agency, public authority, or any other body who, alone or jointly with others, determines the purposes and means of the processing of personal data. The GDPR imposes specific obligations and responsibilities on both data controllers and data processors, and requires them to enter into a written contract or other legal act that sets out the subject matter, duration, nature, and purpose of the processing, as well as the obligations and rights of the data controller.
In this scenario, BHealthy is the data controller, as it determines the purpose and means of collecting and sharing its customer information with Natural Insight. Natural Insight is the data processor, as it processes the customer information on behalf of BHealthy for the purpose of determining the price point for BHealthy's new sunscreens. However, Natural Insight also intends to use the customer information for its own purpose of improving its algorithms, which may not be aligned with BHealthy's purpose or instructions. This may constitute a breach of the data processing contract and the GDPR, as the data processor must only process the personal data on documented instructions from the data controller, unless required to do so by EU or member state law (Article 28(3)(a) of the GDPR).
Therefore, the only case in which Natural Insight's use of BHealthy's data for improvement of its algorithms would be considered data processor activity is if Natural Insight receives express contractual instructions from BHealthy to use its data for improving its algorithms. This would mean that BHealthy has given its consent and authorization for Natural Insight to process the data for that specific purpose, and that Natural Insight is acting in accordance with BHealthy's instructions. In this case, Natural Insight would still be bound by the data processing contract and the GDPR, and would have to comply with the other obligations and requirements of a data processor, such as ensuring the security of the data, respecting the conditions for engaging another processor, assisting the data controller in ensuring compliance with the GDPR, and deleting or returning the data to the data controller after the end of the service.
The other options are not valid cases for data processor activity, as they do not involve the data controller's instructions or consent. If Natural Insight uses BHealthy's data for improving price point predictions only for BHealthy, it may still be processing the data for a different purpose than the one for which it was collected and shared, and without BHealthy's knowledge or approval. If Natural Insight agrees to be fully liable for its use of BHealthy's customer information in its product improvement activities, it may still be violating the data processing contract and the GDPR, as it is not acting on behalf of the data controller, but for its own benefit. If Natural Insight satisfies the transparency requirement by notifying BHealthy's customers of its plans to use their information for its product improvement activities, it may still be infringing the data controller's rights and obligations, as it is not the data controller's role to inform the data subjects of the processing activities, and it may not have a lawful basis for processing the data for its own purpose.
Reference:
GDPR
Data Controllers and Processors - GDPR EU
Who does the UK GDPR apply to? | ICO
What Activities Count as Processing Under the GDPR?
What constitutes data processing? - European Commission