Note:  Again, there is a fundamental difference in how the רמב"ם and other ראשונים learn פשט in this משנה.  Most say that a קרון is a wagon, and the סיפא of the שלישית tied to straps is a completely different case, where a third animal is tied to a team that is not כלאיםby themselves.  However, the תוספות אנשי שם points out that this is problematic, as this third animal could not pull the load on its own, and the team it is connected with can, so it is not really doing anything ("זה יכול וזה אינו יכול הו"ל מסייע שאין בו ממש").  Perhaps for this reason, the רמב"ם learns that a קרון is composed of three wagons tied together - it is not clear, but it appears that he has two side-by-side, with a third, that is the שלישית of our משנה, behind and between them, tied to the straps that connect the front two.  According to the רמב"ם, the רישא is dealing with a passenger in one of the front wagons, and serves to illustrate the מחלוקת of R' Meir and the ת"ק; the סיפא tells us that the ת"ק holds that it is even forbidden to ride in this third wagon.

המנהיג - While the מלאכת שלמה says that the driver need not even be sitting in the wagon to be liable, the משנה ראשונה notes that as to whether one would be liable for merely driving the animals with one's voice is a there is מחלוקת between ריש לקיש and רב יוחנן in ב"מ.

היושב - Although this person does not do anything to drive the animals, they can sense a passenger's weight causing them to pull the wagon.

אין קושרין את הסוס - In this case the wagon is being pulled by a team that is not אסור by itself, but would be with a horse.  The רע"ב says that horse is tied to the wagon to teach it how to pull a wagon, and this arrangement is forbidden because the horse may occasionally push the wagon with the team hitched in front. 

ולא את הלובדקיס - According to the משנה ראשונה, the חדוש here is that this animal is not a type of camel, and so may not be teamed with camels; תפארת ישראל says that camels are so much larger and stronger, one might think that the relatively diminutive הלובדקיס is not doing anything, קמ"ל.