Basically you are fine except for one major ingredient: you need what amounts to an entire new step in between your steps 3 and 4. You need to have a notion of tangent space and connection between infinitesimally near points before you can proceed to step 4 for an important reason: mathematically, this reason takes the form: the notion of tensor isn't defined on nearby points, but on what a mathematician calls vectors in the tangent space, and physically: because the distance between two nearby points depends, at first glance, on the path you take between them. The point is that logically one starts with the notion of a quadratic differential form, and to go from that to distance you take a line integral over a path from one nearby point to the other. Since the conceptions are logically distinct, and physically distinct, you need to have this intermediate step that I am about to explain:
Step 3 and a half. Tangent vectors. Note: you have to do step 5 before you do step 4 or my new step 3 and a half.
You have to get the notion of a flat vector starting at any point $P$ of your set but heading straight off the set, into space if necessary and you think of your set as embedded in space, or living in a whole world of its own which only has $P$ as a point of contact with your set, if you do not think of your set as embedded in a larger flat space. This notion can be provided by the partial derivatives in different directions of your coordinates, I omit the details unless you want them. Hence for each coordinate function $x^i$ around $P$ you have constructed $\partial \over \partial x^i$, which is a tangent vector, and there are $n$ of these where $n$ is the dimension of your set or space.
The $\partial \over \partial x^i$ for $i$ from one to $n$ span a vector space, the tangent space, denoted $T_P$, since they can be combined to form other linear differential operators on the space of differentiable function on your set.
Only now can you define the metric tensor field: it is an association (for each $P$) of a positive definite quadratic form on each $T_P,$ which is differentiable as $P$ varies.
I.e., before you can, as you put it,
Decide on a metric that gives the infinitesimal distance between two adjacent points.
you have to say that two adjacent points determine one of these tangent vectors that give the direction you would head off in in order to get from one of them, $P$, to the other one. I.e., you really have to replace the notion of « adjacent points » by « vector in the tangent space,» and « infinitesimal distance » by « Euclidean distance inside the tangent space.»
And by now, perhaps you see that at least part of your step 6 really has to be done before you can pick a metric tensor field since it is, after all, a tensor field...
But except for getting things in the wrong order and leaving out an essential step, you were fine.
Well, as already pointed out by somebody, your matrix had better always be positive definite symmetric. But what I want to further point out is that, technically,
Assuming I do all the above with a flat plane (a table top for example),
using Cartesian coordinates.
well, umm, all the above included picking a metric tensor field, right? so you already picked it. Not to be picky, but what you really want to ask is, alright, assuming you have all the notions of differentiable manifold, coordinates, tangent vectors, tensors, vector fields, tensor fields, now what conditions do four functions of two variables have to satisfy so that they give the components in Cartesian coordinates of a metric tensor field? And as pointed out, you need that they define a symmetric positive definite matrix for all values of $x,y$ when you arrange them in a matrix like you did.