OK, now it's more clear what he meant. He did not mean that "level-headed ketch" is an actual term, he meant that it's a ketch
which happens to have a mizzen equal in height to the mainmast.
His argument is that the forward boom is longer than the after one, so the forward-most mast must be the mainmast. There is something to that, I think.
If you accept that idea -- that it can't be a schooner because the forward-most mast carries more sail -- then, you could also call it "a weird ketch
with an unusually big mizzen".
By this argument, your rig could even be a ketch if the after-most mast were taller than the forward mast -- so long as the boom were longer and sail area greater. That would be even weirder -- "a schooner-like rig which is however actually a ketch, since the forward-most mast is the mainmast."
I would probably stick to conventional terminology, myself, and call it "a schooner rig which is, however, ketch-like since the foremast carries more sail than the mainmast". For most people, ketch or schooner will be defined by the relative heights of the masts.
For you, the main thing really will be how do you call your masts. Is the aftermost mast the mainmast, or the mizzen? I would probably call it the mainmast.
As to the position of the after-most mast -- I disagree with your rigger. Your rig could never be a yawl, because a yawl is only arbitrarily, by old racing
rules, defined as a two-masted vessel with mizzen behind the rudder
post. In essence, a yawl is a two-masted vessel with a mizzen sail so small that it is good really only for balance, not intended as a significant source of drive, and the position of the mast is not actually the main thing. Your boat could never be that, even if the after-most mast were behind the rudder
Tayana made quite a few schooners, mostly 55/58 feet. The rig, like yours, has equal height masts. Tayana calls them "staysail schooners". Your rig looks like the same thing to me. Do you have a staysail? A wonderful sail which is perhaps the best thing about a split rig. The other cool sail you can use with a split rig is a fisherman.