How is the One Time Pad (OTP) perfectly secure?
You've actually been trapped by the mindset that OTP will hide all information about the underlying plaintext.

This is not true as you have observed.

The definition of perfect secrecy, given in Introduction to Modern Cryptography by Katz-Lindell, reads like this:


  Definition 2.3 An encryption scheme $(\text{Gen, Enc, Dec})$ with message space $\mathcal M$ is perfectly secret if for every
  probability distribution over $\mathcal M$, every message
  $m\in\mathcal M$, and every ciphertext $c\in \mathcal C$ for which
  $\Pr[C=c]&gt;0$:
  
  $$\Pr[M=m\mid C=c]=\Pr[M=m]$$


In other words, if you have the ciphertext at hand, you won't learn anything new that you didn't already know about the plaintext.

The OTP satisfies this (proved in "Introduction to Modern Cryptography") and thereby is perfectly secret as an encryption scheme. In short it first shows that $\Pr[C=c\mid M=m']=2^{-l}$, then uses this to show that $\Pr[C=c]=2^{-l}$ and concludes via $\Pr[M=m\mid C=c]=Pr[M=m]$ (using Bayes' theorem).