The difficulty with parish councils is that they can often desire an informal meeting where the pomp and circumstance that you would associate with borough and district levels is dispensed in order to make the council more approachable to the public. We only use proposals and resolutions as I find the term 'motion' confusing and unnecessary - after all you are putting forward a suggestion and asking for the council to agree or reject it, that is all.
The agenda has to specify the business to be discussed and that is stated in the LGA 1973 (Schedule 12 I think). There is no requirement to mention who the proposer is as it is the Clerk who creates the agenda for the meeting. That said, if the proposal came from a particular committee, I would mention that fact to provide an audit trail.
The wording on the agenda, in my practice, needs to state what the decision will be on but it is often the case that the councillors will opt to amend the resolution following a debate OR there may be an item for the council to decide between 2 courses of action, or a number of suppliers, or, heaven forbid, the clerk is asking the council to actually decide itself, rather than being spoon fed by the Clerk. In this instance, you would not be able to put an exact resolution on the agenda but it will be recorded in the minutes.
It might be worth pointing out that the agenda is not a legal record of the meeting, the minutes are. The agenda just sets out the business to be discussed.
During the debate the exact wording of the resolution will be agreed and it will be that which is voted on, after being proposed and seconded.
An example of all this is in a forth coming meeting that we are holding via Zoom. There is an agenda item to discuss a lease that was in process before the lockdown. I am unable to put on the agenda "To approve the signing of the lease" as I don't know what the council want to do as a next step. Instead I will be putting "To receive an update on the lease and to agree next steps". That is a more accurate description of what will be taking place.
Finally, there is no justification to make up resolutions on the night. There has to be an agenda item forming the basis for the resolution and if there isn't it should be carried over to the next agenda.
Hope the above helps.