Google-sheets – Show a Running Target Based on Today

The command offset does what you want: =offset(D3, -1, -2) is the value 1 row above and two columns to the left of D3. So, your formula for D3 could be

 =offset(D3,-1,0) + offset(D3,0,-2) - offset(D3,0,-1)

Extended down the column, it will do the intended computations regardless of row insertions.

It's easier to write a custom function for this than to look for a MMULT solution that may not even exist. The function would be used as

=runningtotal(B2:B12, A2:A12, "New")

where B2:B12 is the range with data, A2:A12 is the range with conditions for restart, and "New" is the value that tells that the total should restart.

The script runs a loop over the data, aggregates the total in total, and restarts when needed.

function runningTotal(data, conditions, str) {
  var output = [];
  var total = 0;
  for (var i in data) {
    total = (conditions[i][0] == str ? data[i][0] : total + data[i][0]);
    output.push([total]);
  }
  return output;
}