Just wanted to follow up to say that the problem seems to have been solved.

1) DrByte was very correct in that the default "Offline Processing" credit cart module does produce the credit card form on the checkout payment page. While the "secure payment" field on that same page was associated with our merchant account (Transfirst).

2) The problem, and it was overlooked by many including me, was one of the settings for the Transfirst/Transaction Central module in Admin > Modules > Payment

The Gateway Method .... I had it was TFHosted. The other method was HTTPS Post. Since these websites have an SSL certificate, the method should be the HTTPS Post.

Once I switched that over and made sure the "Offline Processing" module was uninstalled - I got the correct credit card form on the checkout payment page. And while on the phone with Transfirst Integration Dept, I did a test purchase using a fake credit card info and while that produced an error for me as a customer, it did product an order in the zencart admin AND Transfirst immediately saw the transaction on their end (with the appropriate error msg and such).

We'll test a few more times - but thus far, it works!