真礼 and 土下座—bowing or prostrating oneself?

In lessons in chanoyu 茶の湯, Tea ceremony, we've done three kinds of bow (rei 礼), aligning with the shin-gyou-sou hierarchy of formality:

• the shin (真, "true"?) bow, the most formal, for entering and exiting the space, etc
• the gyou (行) "moving" bow, about halfway down, as for passing tea, or 道具 Tea implements for perusal, to the next guest
• the sou (草) "grass" bow, just touching the mat and nodding as an acknowledgment (eg, while passing food), especially when you need things to keep moving

I've only heard these terms, so I can't be sure; the bow on entering and leaving class itself may in fact be a 師礼 (shirei), a bow to the teacher, though how that would differ from shin, I'm not sure.

I gather the shin-gyou-sou paradigm applies to other contexts in Japanese culture; certainly it's there in calligraphy (書道)—kaisho 楷書 as, though maybe not most formal, standard; 行書 as the more fluid "cursive script"; 草書 as writing so stylized and/or introspective as often to be illegible.

Recently I've learned of the phrase dogeza 土下座, which is literally down-ground-position but can be translated as "prostrating oneself"; in practice it seems similar to the shin 真 bow. How are these different? Are they?

In Tea class, the shin bow is done from seiza (kneeling), hands fully down on the mat at chest level, fingers and thumb together, elbows out, nose almost to the mat. It's measured and respectful in Tea, though I guess not always in other contexts:

To me, the English phrase to prostrate oneself means something else entirely: It's a body position derived from Catholicism, expressing total subjugation, repentance, humility. Body face-down on the floor, feet together, arms either at the sides or out at a right angle, to form a cross:

But a search turns up far more instances of 土下座 as prostration, so I must just be thinking of it from my own background.