Checked my Comm 10, my Comm 10A and my early Comm 12 (all yellow tractors) this AM and none of them have the K-series designation on them. The engines on the two 10HP tractors are designated model #G-24 and on the 12HP machine, model #G-30-E. I forgot to look at my red colored Comm 12 this morning (later vintage - 1974), but I believe that motor does have a K-series designator on it.
If you need them, the spec # on the Comm 10 was 46341D and on the Comm 10A, #46520D. These engines would be equivalent to the later K241 designation and the ones I've observed had an "S" suffix on them, as have the walk-behind 12HP tractors (K301S). Keep in mind that the 10HP and 12HP tractors from that era share the same engine block - with the only difference being the 12HP having a slightly larger cylinder bore.
My 1974 Comm 12 does have the K-series designation (K301S). Obviously my late 60's Kohler powered machines do not. Hopefully that information will provide some guidance. I haven't heard of the "T" engine designation - even on the rider I have, the engine is designated as K301S with a different spec number (probably equating to the different carb linkage, fuel line arrangement and exhaust routing needed for the riding tractors).
Thanks for the info - does your Comm 10 have a magneto ignition behind the flywheel or a coil mounted outside the engine? Mine has the magneto ignition and I'm wondering if it's original or someone changed the engine/ignition system.
Both of the Comm 10's appear to be magneto fired - looks like the early Comm 12 has an external coil assembly on the left side of the engine (from operator's position). The later Comm 12 (red) has the standard coil assembly on the right side of the motor.