Rambam writes in Hilchos Melachim (11:1):
...anyone who does not believe in (the Messiah), or one who does not
wait for his arrival - he is not merely denying the other prophets but
rather denies the Torah and Moshe Rabbeinu.
Moreover, the original source for these fundamental beliefs as described by the Rambam come from his introduction to Perek Chelek in a more straightforward fashion:
The Twelfth Fundamental Principle refers to the Messianic Era. We are
to believe as fact that the messiah will come and not consider him
late. If he delays, wait for him (Hab. 2:3); set no time limit for his
coming
From these sources we see that there is a two-fold element to belief in the ultimate redemption: 1) to believe that there is an anointed Redeemer and that 2) we must actively wait for his inevitable arrival. Those two elements are distinct and must be met for one to fulfill that principle of faith.
Thus, the formulation rendered by the anonymous author of Ani Ma'amin (as noted in a now deleted comment to the OP's original question [before the question was edited to reflect the fact that Rambam did not in fact write the piece that is printed in most siddurim]) is able to address these two crtierion by a) asserting the belief in an anointed messiah (I believe in the arrival of the mashiach) and b) asserting the belief that he can come at any day, at any time and that we must actively anticipate this arrival (even though he may tarry... I await his arrival every day).
The anonymous author's poetic license seems fitting to me to cover both aspects of this fundamental of faith as it speaks both to the necessity of the latter part of the principle while accounting for the very reason why one may struggle with that aspect of the principle (i.e, the perception that there is a delay in his arrival).
