Hi OP the police should have told you the offence that is being investigated, it might be Malicious Communications for this but without knowing more I couldn't say.
If the suspect refuses the voluntary then yes they would be arrested, but in situations such as this where you know who the suspect is/where they live etc it is common practice to offer them the chance to come and give their side without the dramatics of turning up and taking them in a police car to custody where the sergeant may not authorise their detention anyway as they will ask why you haven't first tried the voluntary attendance route first.
We do this when people are reporting an offence after it has happened, I deal with cases for harassment/stalking/mal comms/criminal damage all the time where I invite the suspect in to the station rather than going out and nicking them as that is much more resource intensive and a longer process.
When they attend they will be offered a solicitor and placed under caution we check their welfare/health needs and if they are fit to be legally interviewed, for instance are they sober etc, have they had their medication if they need it.
Then they are placed under caution 'you do not have to say anything,....'
And questioned just as they would be in a custody interview.
They are warned as everyone that if they choose not to comment this could have an adverse affect in future should the matter progress to court proceedings.
It is recorded on a digital machine. Then at the end they are free to go and then what is called a 'disposal decision' is made usually after they have left.
This means what we are going to do with them, so is there enough evidence to charge them with the offence, or we need to further investigate.
Depending on a person's history of offending there are several outcomes and you'd need to talk to the officer in your case about what these might be.
Hope this helps a bit.