The standard way to define an ideal is as follows:
$I$ is an ideal if it satisfies the following conditions:
$(I,+)$ is a subgroup of $(R,+)$
$\forall x \in I$, $\forall r \in R :\quad x \cdot r \in I$
$\forall x \in I$, $\forall r \in R : \quad r \cdot x \in I$.
A potentially more geometric way of looking at it is to define it as follows:
"An ideal in an algebra is a subset of the algebra that is also a module over the algebra"
V.I. Arnold: Singularities of Differentiable Maps, Vol. 1 - Introduction
(Intuitively, it's more or less just a vector space over the algebra)
Is there a geometric interpretation to accompany this perspective, i.e. a nice picture summing the concept up?
For example, for quotient groups there is a geometric picture that sums up the entire concept:
(Dummit and Foote: Abstract Algebra - Chapter 3)
Similar pictures can be provided for a lot of the concepts in group theory - how about ring theory?
I picture an ideal as follows:
The ring is sketched by the coordinate axes going off in two directions. It only goes off up and to the right because in this picture my ring has a unique unit which is the origin. Multiplying two elements in the ring is adding them as vectors. Thus multiplication takes you more to the top right. The ideal is the shaded region. It has three generators and consists of all elements that are above or to the right of a generator.
This picture is quite accurate when I is monomial ideal in a polynomial ring, that is generated by monomials. It is far off in all other cases, but the question was for a picture that captures something, and your picture of the quotient group does not capture the word "group" either.
The important thing is to use these pictures only as mnemonics in the beginning, and not fool yourself into thinking that pictures can replace working formally.