RSA cipher: ambiguous or break by eth root? Confused about RSA message cipher: $c = m^e (mod\ n)$

So if $m^e$ can be greater than n, then you can get duplicate ciphers for the same message.   Most postings I've read say that $m^e$ must always be smaller than n to counteract.

But if $m^e$ is smaller than n, then $c = m^e$ and can just be decrypted by $\sqrt[e]{c}$.  Most postings I've read said that $m^e$ should be greater than n to counteract.

So it seems like there are only 2, differently broken, options.  How is this reconciled in practice?
(My assumption is that "padding" reconciles this, but I'm not sure how.)
Due to poor use of notation, the question uses an incorrect (or at least incomplete) definition of textbook RSA encryption, which actually defines ciphertext $c$ for a given $m$ as:$$c=m^e\bmod n$$This can be read as: “ $c$ equals [pause] $m$ to the $e$th power reduced modulo $n$ ” where power or/and reduced can be left implied, and the pause emphasizes that $c$ is the result of the final modular reduction. It means that $c$ is the remainder of the Euclidean division of $m^e$ by $n$, which unambiguously defines $c$ for given $m\ge0$, $e&gt;0$, and $n&gt;0$. It equivalently means that $0\le c&lt;n$ and $c\equiv m^e\pmod n$, as defined below. 

The notation $c\equiv m^e\pmod n$ can be read as: “ $c$ is equivalent to $m$ to the $e$th power [pause] modulo $n$ ” where the pause emphasizes that modulo pairs with the earlier equivalent, rather than with an implied reduction. For non-zero $n$, it means that $m^e-c$ is a multiple of $n$. There are infinitely many $c$ matching this.

Notation rule: when $\mod\;$ is without an opening parenthesis immediately on its left, and without an $\;\equiv\;$ sign anywhere on its left, it means reduction modulo, or equivalently remainder. Otherwise, it tends to mean equivalence modulo. The later is unambiguous when $\mod\;$ is with an opening parenthesis immediately on its left, and with an $\;\equiv\;$ sign somewhere on its left.



Indeed, in order to avoid guess of $m$ as $\sqrt[e]c$, it is necessary that $m^e\ge n$ holds, at least with overwhelming odds. The recommendable method to ensure this is not to directly avoid small $m$; rather, it is to pick $m$ essentially as a random integer with $0\le m&lt;n$. It is safer, and practiced, to choose $m$ in the later way, and then use $m$ as a key to a symmetric cipher protecting the confidentiality of the actual message. Other methods exist that allow conveying with $c$ a small message $M$ reversibly turned into an integer $m$ that is random enough, see PKCS#1 encryption schemes.