Circular slot is what type of loading?

Some Vivaldi antennas use a circular slot at the end of the slotline while some don't. I know that a simple rectangular slot represent an inductive loading. But what type of loading is a circular slot that ends on a rectangular slotline?

Hi,
This kind of structure is called a Marchand Balun.
In a Marchand balun, the microstrip and the slot line meet in orthogonal directions on
the opposite sides of the substrate. Microstrip line ground plane is in this case created
by one side of the slot line metallization. Microstrip line is terminated by a stub, which
creates a virtual short at the point of crossing, virtually shunting the microstrip to the
other side of slot line metallization. That enables the propagating field to couple into the
slot line on the opposite metallization layer. As the slot line is terminated by an open
end at the point of transition, the field can propagate through such transition without
any reflection and insertion losses (in an ideal case).
There are variations of this kind of microstrip to slotline transitions (with microstrip radial stub, with microstrip shorted with a via, double Y balun which is a combination of open and short).

So since at the 'cross over point' there is a virtual open on the slot side of the substrate, there's a short at the end of the slot right? So that means the circular slot is an inductive loading, right?

First the virtual short is on the microstrip side and not on the slotline side. It is necessary to create a current node to couple to the slot.
Then picture you sent does not really reflect the reality of practical Vivaldi antennas, according to my humble experience too.
Normally the circular slot is by far closer to the "horn" mouth and acts as a broadband open. The slotline of your interest should be seen as a very short-length line for open-broadband tuning purpose.
But to answer your question, strictly regarding your picture and forgetting the Vivaldi feeding aspect, that would be an open inductively loaded.

Vivaldi antenna with circular open slot

Ok. That makes sense. Thanks. But I'v found some designs that implement something weird that leads to a short at the horn's mouth.

maybe a short transformed in an open with a lambda/4 tline, but normally not an especially broadband open.
Do you have a picture?