Using a switch would be more useful. And then all you need to do it filter out the objects where they match - no need for an additional loop.

function getCatType(order) {
  switch(order.type) {
    case 'Invoice':
    case 'One-time':
    case 'Free': return 'Once';
    case 'Semester': return 'Recurring';
    default: return null;
  }
}

const paymentsResults = [
  { type: 'Invoice', id: 1 },
  { type: 'Semester', id: 2 },
  { type: 'Free', id: 3 },
  { type: 'One-time', id: 4 }
];

function typeOfPayment(type) {
  return paymentsResults.filter(order => {
    return getCatType(order) === type;
  });
}

console.log(typeOfPayment('Once'));
console.log(typeOfPayment('Recurring'));

You can use reduce and also you can check for the types you need directly in the reduce cb, mix this with a regex and you can extract what ever types of orders you want by creating a simple array with the types you need. Assuming your paymentsResults are something like this array you can do this:

const paymentsResults = [

  {
    order: {
      type: 'Invoice',
      total: 10
    },
    price: 2
  },
  {
    order: {
      type: 'One-time',
      total: 11
    }
  },
  {
    order: {
      type: 'Free',
      total: 12
    }
  },
  {
    order: {
      type: 'Semester',
      total: 13
    }
  },
];

let extract = ['Invoice', 'One-time', 'Free'];
const regex = new RegExp(extract.join('|'), 'i');
let paymentsOnce = paymentsResults.reduce((acc, obj) => {
  if (regex.test(obj.order.type)) acc.push(obj);
  return acc;
}, []);
console.log(paymentsOnce);

I assume you are confused with what filter() is actually supposed to do.

filter() is an Array-Function iterating over the given Array and returning the objects that fit the filter-conditions defined in the callback function.

Therefore you don't need the for-Loop to get into each Item of your array, but simply call your filter-function on the array itself.

This only applies if you have an Array of Objects with the key type in paymentsResults. Sadly you didn't provide your origin data-structure. If it is not like I assumed, then please comment down below, what your data structure looks like.

for (let i = 0; i < paymentsResults.length; i++) {
const paymentsOnce = paymentsResults[i].filter(o => getCatType(o) === "Once");
} 

Will become:

// ONLY GIVEN IF YOUR STRUCTURE LOOKS LIKE 
// paymentsResults = [{type: 'Thing'}, {type: 'Thing'}, ...];
const paymentsOnce = paymentsResults.filter(o => getCatType(o) === "Once");