Vector logo for website

I need to rebrand and my logo needs a complete change, so I want something that looks contemporary but clearly tells people they are looking at a site aimed at people who work in care in the UK- This is vital.
I need it in Vector, jpg,pdf etc , in sizes for social media too.

I will also need the colour name that we agree upon, so that I can match it up with other areas of my wordpress website.
This is the colour i wan to be close to and the logo.
https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/927527/6.6573_DHSC_CARE_Brand_guidelines_Web.pdf - The logo needs to be close in colour and similar, but not the same font( due to copywright)
This is my website https://www.careindustrynews.co.uk/

This is my significant competition: https://www.caremanagementmatters.co.uk/

This is a leading exhibition site that heads my business sector, i have attached their logo too. https://www.caremanagementmatters.co.uk/event/care-show/