Thanks for reaching out.
As a manual tester, you could theoretically position yourself in any of the fields you’ve mentioned. You just have to look for the activity performed by the company you are applying for.
If their objectives relate to creating their own software (or software ordered by third-parties) that automatically performs a number of calculations and produces visualisations, then this sounds like the right place for you to be. Such software can be the focus of a company operating in any of the fields you’ve mentioned – data analysis, data science, BI, or data engineering.
How to switch? – the demand for specialists like you is rising, however it will be your task to find companies that need the help that you can offer. Don’t be afraid to contact potential employers and ask for specific questions regarding their job requirements. Thus, you can know better if the position in question could be right one for you.
Hope this helps.