The GDPR defines profiling as any form of automated processing of personal data consisting of the use of personal data to evaluate certain personal aspects relating to a natural person, in particular to analyse or predict aspects concerning that natural person's performance at work, economic situation, health, personal preferences, interests, reliability, behaviour, location or movements12. Therefore, the relevant factors when determining if a processing activity would be considered profiling are:
whether the processing involves data that is considered personal data;
whether the processing of the data is done through automated means; and whether the processing is used to predict the behavior of data subjects.
The identity of the processor, whether it is the controller or a third-party vendor, is not relevant for the definition of profiling. However, it may have implications for the accountability and responsibility of the parties involved, as well as the data protection rights of the data subjects34. Reference: CIPP/E Certification - International Association of Privacy Professionals, Free CIPP/E Study Guide - International Association of Privacy Professionals, GDPR - EUR-Lex, What is automated individual decision-making and profiling? | ICO, WP29 releases guidelines on profiling under the GDPR, UK: A Guide To GDPR Profiling And Automated Decision-Making - Mondaq

According to the UK GDPR, employees have the right to access and receive a copy of their personal data, and other supplementary information, from their employer. This is known as a data subject access request (DSAR). Employers must respond to a DSAR without delay and within one month of receipt of the request, unless the request is complex or excessive. Employers should perform a reasonable search for the requested information and provide it in an accessible, concise and intelligible format. Employers can only refuse to provide the information if an exemption or restriction applies, or if the request is manifestly unfounded or excessive. Some of the exemptions that may apply in the employment context are: legal privilege, management forecasting, confidential references, negotiations, regulatory functions, and criminal convictions and offences. Employers should disclose the information securely and inform the employee of their rights and the source of the data. Reference:
Right of access | ICO
Subject access request Q and As for employers | ICO
Data Subject Access Request (Employers' Guide) | DavidsonMorris

According to Article 45 of the GDPR, the European Commission has the power to determine whether a third country, a territory or one or more specified sectors within a third country, or an international organisation ensures an adequate level of protection of personal data. This means that personal data can flow from the EU and the EEA to that third country without any further safeguard being necessary. The adequacy decision is based on an assessment of the legal framework, the enforcement mechanisms, the access by public authorities, the international commitments and the cooperation with the EU of the third country or organisation. The European Commission also monitors the functioning of the adequacy decisions and can repeal, amend or suspend them if the level of protection is no longer ensured. The European Commission has so far recognised several countries and organisations as providing adequate protection, such as Japan, Canada, Switzerland, the UK and the EU-US Data Privacy Framework. Reference: GDPR Article 45, Data protection adequacy for non-EU countries, Adequacy decisions | European Data Protection Board

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