Like quite a few Gov.uk pages, that link is badly worded.
What it says is this: "You can apply to either a county court or the High Court if you’re owed between £600 and £5,000".
The way it should be worded is this:
"You can apply to a county court, or to the High Court if you’re owed over £600, for the collection of debts up to £5,000. For debts over £5,000 only the High Court can enforce judgment through the use of bailiffs"
This link goes to the site of a commercial company (which I'm most definitely not advising you to get involved with) but it confirms what I've written:
https://thesheriffsoffice.com/articles/high_court_enforcement_officer_or_county_court_bailiff
Government departments and agencies (such as the Department for Work and Pensions, the Passport Agency, DVLA, HM Courts and Tribunals ,etc) all used to have their
own websites but then some bright spark decided to bring them all together under the 'Gov.uk' banner. Rather than just copying and pasting all of the old content, it was decided to re-write everything in the cause of 'simplification'. Those re-writes appear to have been carried out by people outside the relevant departments, who've no knowledge of the work of those departments. As a consequence, the Gov.uk website is now full of inaccurate and/or contradictory statements. It should only ever be used with extreme caution!