Can a COUNTIF formula "count" when there are multiple values in one cell?

I've created the following formulas to count how many times a person's name is appearing in a configurable (Contact List) column for Project Lead and then Assigned To on another sheet:

=COUNTIF({Project Lead}, Primary@row)

=COUNTIF({Assigned To}, Primary@row)

The Primary@row is directing to the name of the person I have typed out on the sheet I'm creating the stats on (that is the same name being displayed on the sheet I'm referring to).

The formula is working for cells that only list 1 persons name, but where there are cells in the column that include more than 1 name, the formula won't work. Can the COUNTIF or COUNTIFS formula somehow count values when there are more than 1 (and different) values in the same cell? If not, what formula should I be using?

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