There are fine distinctions and not everyone distinguishes in the same way
Head Hunters are people who go and find you someone to meet your specification. This can be a retained search - you get paid regardless of success - or on a contingency - you get paid if you find someone
Recruitment agencies can also be headhunters but can also cover more generic roles and potentially less senior positions. They do can do this as a retained search and selection or by them providing someone from their current database - the fees tend to be lower for the latter
Employment agencies would tend to employ you and then sell your services to the company looking for additional resource. This is primarily for temporary positions.
Job agencies sort can be recruitment agencies or employment agencies.
Lots of people do a mix of the above.
We went for the little used and less understood
term 'recruiters' since they 'recruit' but they recruit as intermediaries rather than as hiring companies or employers or 'end users'.
I have just dashed this note off but over the next week or so I will pull something more definitive together [incuding checking with my US colleagues - who have assured me that the word recruitment translates
]. In the meantime if anyone has a view or an intrepretation - let's hear it.