You can also try:

=COUNTA(FILTER(B:G,A:A="Smith"))

See Also: Count rows with not empty value

Unfortunately, this is functions as designed from Google.
^{Although I'm not sure why it's divergent from the way that Excel calculates COUNTA.}

According to the Google Spreadsheet Documentation on COUNTA:

COUNTA counts all values in a dataset, including those which appear more than once and text values (including zero-length strings and whitespace).

Meaning that the only way to "make [the cells] completely empty" is to delete the entire contents, formula and all. But fear not...

Some workarounds:

1. Hypothetically, you should be able to do this with =COUNTIF(A3:A8,"<>"&""), but Google spreadsheets doesn't support the not equal to operator in the COUNTIF function according to these forums: 1, 2, 3

2. A workaround is to create a truthy or falsy array based on the condition and use SUMPRODUCT to count the truthy values like this:

    =SUMPRODUCT((A3:A8<>"")*1)
   

3. Another option you could pursue would be to write a custom function and add it to Drive.
   It's actually pretty easy to do so. Go to Tools > Script Editor and add the following script:

    /**
     * Works like COUNTA except does not include null strings
     *
     * @param {cellRange} myArray The value or range to consider when counting.
     * @return Returns the a count of the number of values in a dataset.
     * @customfunction
     */
    function CountNotBlank(myArray) {
      var output = 0;
      for (i = 0; i < myArray.length; i++) { 
        if (myArray[i] != "") {
          output += 1
        }
      }
      return output;
    }
   

   Then use like this:

    =CountNotBlank(I31:I)
   

Here is a fairly straightforward approach:

=ArrayFormula(IF(B2:B="",,COUNTIFS(B2:B,"<>",ROW(B2:B),"<="&ROW(B2:B))))

It basically reads "If any cell in Column B is blank, leave the return blank; otherwise count the non-blank entries in Column B including only content in rows less than or equal to the current row number."