A ring is a set $R$ with two binary operations, usually dentoed by $+$ and $\times$, such that $(R,+)$ is an abelian group, $(R,\times)$ is a semigroup (if you like your rings to have identities, then you require it to be a monoid), and such that the two operations are connected via the distributive laws:
$$a\times(b+c) = (a\times b)+(a\times c)\quad\text{and}\quad (x+y)\times z = (x\times z) + (y\times z)\quad\text{for all }a,b,c,x,y,z\in R.$$
So given a ring $(R,+,\times)$, you get a group by "forgetting" about the operation $\times$; and you get a semigroup/monoid by "forgetting" about the operation $+$. In that sense, every ring is also a group (under addition).
It is wrong to say "A ring is a group that is a monoid", because you are not precise enough in refering to the operations, and to the connection between the two operations.
I am also confused by your claim that "rings are used to define groups". How so? A group is defined to be a set $G$ together with an operation $\cdot$ that is associative, has a two-sided identity element, and has two-sided inverses for each element. No notion of ring is harmed in the production of this definition. It is true that many structures that are in fact rings are used to provide examples of groups, but you don't define groups in terms of rings.
Lang's Algebra defines groups on page 7 (Revised 3rd Edition, Springer-Verlag GTM 211) as "a monoid such that for every element $x\in G$ there exists an element $y\in G$ such that $xy=yx=e$. It defines monoid in page 3 as a set $G$ with a "law of composition" (a binary operation) that is associative and has a unit element.
Lang defines "field" on page 84: a field is a commutative ring (the operation $\times$ is associative) in which there is an identity for $\times$, denoted $1$, such that $1\neq 0$ ($0$ is the identity for $+$), and in which every nonzero element has a multiplicative inverse: for every $a$, if $a\neq 0$ then there exists $b$ such that $a\times b = b\times a = 1$. $\mathbb{Z}$ is not a field, because $2$ does not have a multiplicative inverse. It is a ring (in fact, an integral domain).
Common examples of fields are: $\mathbb{Q}$, with the usual addition and multiplication; $\mathbb{R}$ with the usual addition and multiplication; $\mathbb{C}$, with the usual addition and multiplication; $\mathbb{Z}/p\mathbb{Z}$, the integers modulo $p$ with $p$ a prime, using modular addition and modular multiplication.
