there said that 62 000 emails are free,
Where did you get this information?
You can use Amazon SES without any limitations for free if you set it up correctly with your Xenforo installation.
It is worth to pay the 25 USD/monthly for your own IP address though. This is better for sender's reputation. But this is totally up to you and not mandatory.
Your site also has to be hosted on Amazon servers (eg. on their EC2 hosting).
How about if is not hosted on their server? I can't use it?
You can still use it, but then you have to pay $0.10 for every 1000 emails your site sends in a month.
You still can use it. You need to make changes with your webhoster on these specific emails used for sending the emails, for bounces, for complaints etc.
It is too long ago so I do not remember the details anymore. But as far as I remember it was for free. Maybe they changed that now. Check the AWS/SES pages for that
Are you on a shared server / do you have any kind of restrictions regarding amount of emails sent per hour with your webhoster?
Look at the mail addon of dragonbyte. In the newest version he has throtteling in it.
What is this ?
DragonByte Mail is a professional mailing list mod with advanced subscription management features. Additionally, it has a newsletter feature that highlights your most popular threads. Uses DragonByte Mail is the perfect solution for any forum...xenforo.com
DragonByte Mail 4.4.0b1
New Feature: Ability to limit the maximum number of emails per hour