This has to do with ambiguities in the definitions used for certain terms, and software doing strange contortions to avoid breaking backwards compatibility. The USAF's family of Simplified General Perturbations propagators are based on mean element theories of orbital motion. However, there are multiple ways to define and compute mean motion, and historically the Air Force used two of them mixed together.
In the words of Walter, "by mean elements we understand osculating elements from which short-periodic and long-periodic perturbations of the earth’s potential have been subtracted." Brouwer and Kozai both published their formulas for mean elements in 1959, and they had the same goal, but their implementations selected different perturbations to subtract. They were working so simultaneously that their articles appeared in the same issue of the same journal on successive pages. What exactly the differences between them are is very complicated to state, since both papers are mainly just lists of equations, some of which run to 7 or 8 lines each.
The difference that concerns us here is simply that SGP, the original one without the 4 on the end, uses Kozai mean elements; but SGP4, the one used operationally for the last more than 40 years, uses Brouwer mean elements. When that change was made, however, the USAF decided not to change the TLE format at all, so they are still specified in Kozai form even though that means they need to be converted to Brouwer before use. This topic is treated by Vallado, who also discusses ambiguities in the definition of the coordinate system used for TLEs. Furthermore, it points out that conversions between elements, such as mean motion to or from semi-major axis, need to be done using the WGS-72 model that SGP4 used internally; attempting to use numbers from a newer theory produces worse answers, because inconsistency magnifies inaccuracy. The new SGP4-XP, however, uses EGM-96, which is a nice upgrade, but introduces additional differences between the old Type 0 and new Type 4 TLE.
Type 0 TLE contains Kozai elements, as originally used by SGP not 4, is the one in which data has always been distributed. Type 2 TLE contains Brouwer mean elements, as used internally by SGP4, but Type 2 is never broadcast. It exists during calculation, as transformation from Kozai to Brouwer elements is the first thing SGP4 has to do, but it is not retained in that form, unless you save the intermediate result yourself. If you do that, they should be stored as Type 2, which tells the official US government SGP4 not to duplicate the conversion step it would otherwise have started with. If you are using some other implementation not obtained directly from the US Air Force, it may introduce significant error by double-counting.