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 $m\ge0$, $e>0$, and $n>0$. It equivalently means that $0\le c<n$ 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 $0\le m<n$. 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.