Something's not adding up there, Dave. When you invite a member to your country they receive the invite in their email with a confirmation link the user has to click. Once the user clicks the confirmation link in the email there's nothing else you need to do. You shouldn't receive an email back with anything to click, just a confirmation that the player joined.
What I'm guessing has happened is that the folks you sent the invite to didn't click the confirmation link in their email and instead went to your club page and clicked 'Request Membership'. This basically starts the process all over and the email you get is asking you if you'd like to officially invite the player who just requested membership. In that case, there would be a link in your email, however, that's just sending the invite.
Ultimately, the player has to finalize the invitation to join by clicking the confirmation link in the email and I think that's being overlooked.