The word "Avilyn" is simply TDK's own way of adding cobalt to make their own cobalt-doped ferric formulations, then it wasn't used only on the SA but also on superferrics and on other type2 tapes.
For example, the SA-X is still Super Avilyn but with a double coating.
AD-X and AR-X are Avilyn without the Super (I guess because they are tweaked to be used at normal bias range/setting).
Also the AR is some kind of Avilyn... and of course all of their type2 tapes (CDing2, DJ2, etc) are still using a some kind of Super Avilyn formulation.

TDK SA - State-of-the-art Super Avilyn high bias formulation. Delivers unsurpassed performance. High MOL and wide dynamic range.
SA’s advanced cobalt-adsorbed gamma-ferric oxide particle formulation made it the first non-chrome high bias cassette.
High fidelity cassette tape takes a big step nearer perfection with TDK’s revolutionary new Super Avilyn (SA). The special ultra-refined Super Avilyn magnetic particle can be more densely and evenly packed onto the tape surface for an improved powerpacked performance on any cassette deck with high bias (Type II) position. It’s wider frequency range, lower noise
levels and greater maximum output are all achieved with no more head wear than that of normal ferric oxide tape and drastically less than that of CrO2.
Super Avilyn has excellent response in the low and middle frequencies and equals the response of chrome tapes in the high frequencies. It is compatible with all high quality tape machines currently available; where these are fitted with a chrome bias switch, it should be used for optimum results.
SA is also housed in an improved Laboratory Standard Cassette Mechanism; many parts of this super-precision mechanism, such as its double hub clamp and bubble liner sheet have yet to be equalled. This improvement has created a highly precise and stronger mechanism, which effectively reduces any structural warpage and virtually eliminates wow and flutter. Tape to head
alignment is also improved, so tape transport is smoother.

Features of SA:
• Greater coercivity and much finer magnetic particles with greater length- to- width ratio than conventional magnetic materials. 
• Magnetic particles of SA gives personal treatment to each and every signal, thereby helping to yield a clearly defined sound which is crystal clear. It extends the flatfrequency response range from the lowest lows to the highest highs, giving expression to the many sounds and moods of music as they were intended to be heard. It handles exceeding high signal inputs without distortion which is due to increased MOL providing a full dynamic range for both recording and playback.
SA uses the 70 microsecond playback equalization position (High or C-O+) on your deck for a greatly enhanced signal-to-noise ratio and reduced noise level.

SA APPLICATION: For music with great dynamic range, LP dubbing. Record on home component systems. It is the world's most popular high-bias cassette.
SPECIAL FEATURES: The ideal high-bias cassette for all recording applications.