Passing variable to page template with Mod Rewrite

I have a page template in my custom theme which loads a staff directory from a database outside of Word Press. What I am hoping to do is to be able to load information from the directory using their last name in a URL, by passing a variable to my staff directory template through the query string via Mod Rewrite. For example https://www.mysite.com/directory/staff-name/ would load my template https://www.mysite.com/my-template.php?staff_name=staff-name.

In my functions.php file I first add the variable use the add_rewrite_tag function. For example:
add_rewrite_tag('%staff_name%','([^&]+)');

I then set up my rewrite rule using the add_rewrite_rule function. For example (page 201 is the page which uses my directory page template):
add_rewrite_rule('^directory/([\-_0-9A-Za-z]+)/?','index.php?p=201&staff_name=$matches[1]','top');

Lastly in my directory template I attempt to recover the value of the staff name variable from the query string. For example:

$staff_name = $wp_query->query_vars['staff_name'];

Here is what happens when I try to access the directory template via a url:

1. I enter https://www.mysite.com/directory/my-staff-name/.

2. The rewrite rule appears to be recognized, however it appears to forward me to the page with the title of the page in the address bar. For example I end up with https://www.mysite.com/my-page-title/, which is the permalink for p=201 which I put in my rewrite rule.

3. The staff name variable is empty when I try to retrieve it from the query string.

Can anyone give me an idea of what I am doing wrong here? Basically what I would like it to do is when I enter https://www.mysite.com/my-staff-name/, I want this address to stay in the address bar while I am redirected to my directory page template passing it the staff members name through the query string.

Thanks for you help!