Alright let me try explaining this, since it seems to be very confusing to some people, you're not the only one lol.
Let's look at this link, this is the only page that matters by the way, not any other news article, unless it's clearly marked as such.
Most servers are declared standard servers throughout the different data centers. At the bottom of each section you'll find additional newly added servers, and servers that are declared as congested. Now I am not sure why they chose that wording, since it's misleading and causes a lot of misunderstandings, but congested in other words means high population servers.
Leviathan, while during early access has a login queue, and also a character restriction policy on certain days and during prime time, the server is sadly not seen as high population. We might see, now that balmung and gilgamesh closed their transfers and creation, a higher population on levi, but as it stands right now we are not falling into that category just yet.
If we are looking at the bonus initiative and it's requirements, none of them are applicable when it comes to leviathan. To get a free transfer, you have to either transfer off a server that's labeled as high populated, or transfer too an server that's labeled as designated and/or new world.
Example, if I would transfer my character right now to Omega, I would get a free transfer since it's a new world.
I would have to pay for any other transfer that's to other standard worlds within my own or other data centers.
If I take now my alt that's on gilgamesh for example, I could transfer that one to any standard/new or designated world and do not have to pay for it either, since gilgamesh falls under the high population category.
Hope that helps, do also keep an eye on this link listed above, worlds might change depending on the population in and decrease, over the next couple of days/weeks.