Hi J'
This method creates a calendar that extends from the start date to beyond the end date, then uses COUNTIFS to count a specified weekday between those dates.

In the Data Entry table, enter Start Date, End Date and Weekday.
Weekday could be a Pop-Up Menu.
In the Calendar table,
formula in A2 =Data Entry::B1
formula in A3 (and Fill Down) =A2+1
Formula in B2 (and Fill Down) =DAYNAME(WEEKDAY(A2))
(This assumes that Sunday is Day 1 of the week)

Formula in B1 of the Count table =COUNTIFS(Calendar::A,">="&Data Entry::B$1,Calendar::A,"<="&Data Entry::B$2,Calendar::B,"="&Data Entry::B$3)
A quick look at the bottom of the Calendar table to make sure we have sufficient rows to span the start and end dates (I used 1 school term plus a bit):

All good. Sufficient rows to include the end date!
You may want to give the Calendar table extra rows. 366 rows (1 year) should do it!
For a neater appearance, move (cut and paste) the Calendar table to another Sheet. The formulas will still work.

Regards,
Ian.