SQL - LEFT JOIN, but I want COUNT(*) to only count the results from the INNER part of the join

I want to display the number of purchases each customer has made. If they've made 0 purchases, I want to display 0.

Desired Output:

 -------------------------------------
| customer_name | number_of_purchases |
 -------------------------------------
|    Marg       |          0          |
|    Ben        |          1          |
|    Phil       |          4          |
|    Steve      |          0          |
 -------------------------------------

Customer Table:

 -----------------------------
| customer_id | customer_name |
 -----------------------------
|      1      |      Marg     |
|      2      |      Ben      |
|      3      |      Phil     |
|      4      |      Steve    |
 -----------------------------

Purchases Table:

 --------------------------------------------------
| purchase_id | customer_id | purchase_description |
 --------------------------------------------------
|      1      |       2     |     500 Reams        |
|      2      |       3     |     6 Toners         |
|      3      |       3     |     20 Staplers      |
|      4      |       3     |     2 Copiers        |
|      5      |       3     |     9 Name Plaques   |
 --------------------------------------------------

My current query is as follows:

SELECT customer_name, COUNT(*) AS number_of_purchaes 
FROM customer 
LEFT JOIN purchases ON customer.customer_id = purchases.customer_id 
GROUP BY customer.customer_id

However, since it's a LEFT JOIN, the query results in rows for customers with no purchases, which makes them part of the COUNT(*). In other words, customers who've made 0 purchases are displayed as having made 1 purchase, like so:

LEFT JOIN Output:

 -------------------------------------
| customer_name | number_of_purchases |
 -------------------------------------
|    Marg       |          1          |
|    Ben        |          1          |
|    Phil       |          4          |
|    Steve      |          1          |
 -------------------------------------

I've also tried an INNER JOIN, but that results in customers with 0 purchases not showing at all:

INNER JOIN Output:

 -------------------------------------
| customer_name | number_of_purchases |
 -------------------------------------
|    Ben        |          1          |
|    Phil       |          4          |
 -------------------------------------

How could I achieve my