What if the core for a toroid was not a closed loop?  Could you then measure the current and factor in the "lossy" state of the core?

I have a testbed or 4 that are setup this way and there is a field emanating from it as well as one can be introduced as such to generate an output from the winding, so I am thinking this is not a "valid" way to test."

I strongly disagree. 

There are fields in that open area.

Lets take the toroid diagram in the vid and make it a transformer. Primary on the left and secondary on the right. Now, if the primary induces a field in the core, how does the secondary get induced if the field is only in the core?    

The field of the primary has to 'cut' the windings of the secondary.

If you put a wire through the very center of the toroid opening,not even touching any toroid windings or the core, you will get current induced into that wire from an input to the toroidal coil. So how can that be, if the field is only in the core? ?

Mags,
Correct, but it is not a magnetic field.
It's quite easy to understand how, but you first have to understand that a changing flux (B-field) creates a changing electric E-field, which is at 90º. Second you need to be aware that the E-field is present both inside and outside the toroid. Third, you need to understand that it is actually the E-field which is responsible for both self and mutual induction. Fourth, the B-field inside the core is conducted from the primary through to the secondary due to the high permeability core.
When using a toroid core, no it is not required. What is required is that the flux created in the core by the primary, must go through the center of the windings of the secondary, the same way you can make a transformer by placing two coils side-by-side on a rod core.
It is not only normal for this to be the result, but it is required. It must occur. But as I mentioned, you must be aware of and carefully consider the E-field in this interaction. When you see this, it is simple and obvious.

Ask questions if you are still not clear on what I'm saying.

Mags,
Correct, but it is not a magnetic field.
It's quite easy to understand how, but you first have to understand that a changing flux (B-field) creates a changing electric E-field, which is at 90º. Second you need to be aware that the E-field is present both inside and outside the toroid. Third, you need to understand that it is actually the E-field which is responsible for both self and mutual induction. Fourth, the B-field inside the core is conducted from the primary through to the secondary due to the high permeability core.
When using a toroid core, no it is not required. What is required is that the flux created in the core by the primary, must go through the center of the windings of the secondary, the same way you can make a transformer by placing two coils side-by-side on a rod core.
It is not only normal for this to be the result, but it is required. It must occur. But as I mentioned, you must be aware of and carefully consider the E-field in this interaction. When you see this, it is simple and obvious.

Ask questions if you are still not clear on what I'm saying.