You cannot "press charges", that's a myth. You can support a prosecution, but its not up to you to decide who gets charged or arrested. Most prosecutions will struggle without the victims support in terms of a statement, but some offences can be dealt with without the support of the victim.
Sometimes, like above, you're input and opinion will be asked for in terms of disposal methods, but the final decision still rests with police/cps.
Domestic violence is a good example, 9 times out of 10, the victim won't assist the police (even after ringing up themselves), but it won't stop the offender being arrested, and possibly charged if other evidence is available.
The concept is still called "pressing charges" but what most people don't understand is that pressing charges is not the same as prosecution. And as you say, it is up to the CPS whether they decide to prosecute.