These two posts from 2004 provide some helpful guidance on crafting the Prayers of the People in a liturgical worship setting. Published by the Catholic diocese of San Jose, the posts describe these prayers as General Intercessions or the Prayers of the Faithful.
- How to write General Intercessions (A step-by-step process)
- 10 Principles for Writing General Intercessions (Basic principles)
The author cites the instructions to the Roman Missal then in effect, and I would agree with it completely.
In the Prayer of the Faithful, the people respond in a certain way to the word of God which they have welcomed in faith and, exercising the office of their baptismal priesthood, offer prayers to God for the salvation of all.