Need an explanation of what e-mails are sent when

I'm having trouble configuring amember 4 to send a thank you e-mail to members when they join or renew. It sounds simple, but it seems there are at least 4 places to set this up, and none of them are working the way I would like. I see under Set-up/configuration::e-mail:
1) Send registration e-mail - this appears to be triggered if someone registers, regardless of whether they've paid. I don't want to send an e-mail unless payment is received.
2) Send sign-up e-mail - this appears to only send an e-mail if the person is subscribing for the first time. I want to send a thank you e-mail whether it's a completely new member or someone renewing their membership.
3) E-mail payment receipt to user - I could edit the template here to send a "thanks for being a member" e-mail, but this does not appear to send an e-mail when the payment is added manually. We have some people mail in their membership dues, and it would be nice to have the welcome e-mail sent automatically whether the payment is manual or online.
4)Under Protect Content:: e-mail messages, there is the option to set up an auto responder "immediately after a subscription is started". I had this option enabled, but we had a member renew her membership 3 months early, and she did not get a thank you e-mail. Possibly this is because her add-on membership doesn't start until her previous one expires in September? When I send a test e-mail, it arrives fine so I don't think the problem is with my basic e-mail configuration.

If someone could supply a detailed explanation of what triggers each of these e-mail options to be sent, or what I am doing wrong, I would really appreciate it.

We recently upgraded to amember 4. If I go to edit user:: payments:: select a product:: add access manually, it updates the status as active, but does not generate an e-mail. I did a test and did not receive an e-mail, nor is it listed in the mail queue log. Is there another way to add payment manually?