Your video doesn't load. I can tell you that there is no way a private ticket will be saved as public unless you have a category which is set to public-only tickets (Allow Only is set to Public Tickets) or you have a user who does not have the Create Private privilege. In these cases even if you select the ticket to be private it will end up being public. In the first case because the category is set up to only have public tickets. In the second case because the user is set up to only have public tickets.
Most likely your user belongs in many groups. One of these groups has the Private Tickets privilege set to Deny. Remember that Joomla! considers Deny stronger than Allow. Therefore, even if the user belongs in another group that has the Private Tickets privilege set to Allow, the effective privilege is Deny i.e. they are not allowed to file private tickets. This is how Joomla! works, it's not a bug.
You should NOT use the Deny setting in privileges unless you really know what you are doing. You should be using the inherited which is, by default, a "soft deny" i.e. the user does not have the privilege but if you use Allow you give it to them. There are some really good links on Joomla ACLs in our documentation. Please read them to understand how privileges work.
Nicholas K. Dionysopoulos
Lead Developer and Director