Is there more than one definition of homotopic equivalence for two spaces? Is there more than one definition of homotopic equivalence?
Wolfram.com under the topic Homotopic says the unit circle and a point are homotopic (presumably meaning homotopic equivalent) in the plane.  Wolfram also says one must define the “ambient space” to determine whether two objects are homotopic or not.
The Wolfram site goes on to say that there are other ways to compare two spaces via homotopy without ambient spaces. Then an alternative definition that matches Hatcher’s definition (p3) is given.
So, we have two different definitions of homotopic equivalence, one that results in the circle and the point being homotopic equivalent, and the other resulting in the circle and the point not being homotopic equivalent.
Am I understanding this correctly?  Is the key in the nomenclature “homotopic” vs. homotopic equivalent”?
 A: We need to be careful with the terminology. I am following the definitions of Hatcher.
Two continous maps $f,g\colon X\to Y$ are homotopic, if there is a homotopy $$H\colon X\times[0,1] \to Y$$ such that $$H(\cdot, 0) = f,\ H(\cdot, 1) = g.$$
A map $f\colon X\to Y$ is called a homotopy equivalence, if there is a map $g:Y\to X$ such that $$g\circ f \simeq \operatorname{id}_X$$
$$f\circ g \simeq \operatorname{id}_Y$$
Now if there is a homotopy equivalence $f$ between the spaces $X,Y$, they are called homotopy equivalent.
Now $\mathbb{R}^2$ is simply connected, thus the unit circle $S^1\subset \mathbb{R}^2$ embedded as the image of a map  $$f:S^1 \to \mathbb{R}^2$$ is indeed homotopic to the constant map. Indeed, this observation relies on the ambient space $\mathbb{R}^2$ being simply connected.
Removing a point, say the origin  $p = \mathbf{0}$ from $\mathbb{R}^2$ violates the simple connectedness of $\mathbb{R}^2$, and therefore the inclusion of the unit circle $S^1$ in $\mathbb{R}^2$ is no longer homotopic to the constant map.
I've looked up the definitions provided by wolfram alpha and when they call two subspaces being homotopic, i think they mean that there is a homotopy $H$ between two maps $f,g\colon X\to Y$ and consider the respective images of these maps, i.e. $f(X), g(X)\subset Y$ as subspaces being homotopic.
If there is any notion about (sub)spaces being homotopic, it's most likely all about their respective maps being homotopic. That is for example the inclusion of a subspace (as for the example of $S^1 \hookrightarrow \mathbb{R}^2$ being homotopic to the constant map).
