From the comments I think the theorem you are looking for is this. I'll be a little fast and loose just to make it easier to state.

Let $M$ be a manifold with corners and $N$ a submanifold, potentially not properly embedded. So $[0,1]^2$ as a subset of $\mathbb R^2$ would qualify, for example, and similarly $[0,1]^2 \times \mathbb R$ as a subset of $\mathbb R^3$ or $[0,1] \times \mathbb R^2$.

Then there exists an essentially unique regular neighbourhood of $N$ in $M$ and it has a description similar to a stratified space, all definable in terms of the normal bundle of $N$ in $M$, and the stratifications of $M$ and $N$ respectively.

The `top' stratum is the normal bundle of $N$ in $M$.

The co-dimension $1$ stratum consists of the restriction of the normal bundle to $M$ in $N$ to $\partial_1 M \setminus \partial_1 N$, but then you have to take a direct sum with $[0,\infty)$. Here $\partial_1 M$ means the co-dimension one boundary stratum of $M$. Similarly $\partial_2 M$ means the co-dimension two strata, i.e. the traditional `corners', for example $\partial_2 [0,\infty)^2 = \{(0,0)\}$.

The co-dimension 2 stratum of the regular neighbourhood consists of the normal bundle to $M$ in $N$ restricted to $\partial_2 M$, but then you have to take a direct sum with $[0,\infty)^k$ depending on which stratum of $M$ you are in. $k=2$ for the interior stratum, $k=1$ for the co-dimension one stratum and $k=0$ for the co-dimension two stratum.

etc.

These spaces you glue together in the natural way, and you have a corresponding embedding from this total space to the ambient manifold $M$.

I could revise this to be more precise but let me know if that makes sense. It corresponds to your images, but my response deals with some cases you did not include in your images.

From the comments I think the theorem you are looking for is this. I'll be a little fast and loose just to make it easier to state.

Let $M$ be a manifold with corners and $N$ a submanifold, potentially not properly embedded. So $[0,1]^2$ as a subset of $\mathbb R^2$ would qualify, for example, and similarly $[0,1]^2 \times \mathbb R$ as a subset of $\mathbb R^3$ or $[0,1] \times \mathbb R^2$. I suppose this should be called something like semi-properly embedded. We do demand that each connected stratum of $N$ is mapped to just one connected stratum of $M$, as in the above listed examples.

Then there exists an essentially unique regular neighbourhood of $N$ in $M$ and it has a description similar to a stratified space, all definable in terms of the normal bundle of $N$ in $M$, and the stratifications of $M$ and $N$ respectively.

The `top' stratum is the normal bundle of $N$ in $M$.

The co-dimension $1$ stratum consists of the restriction of the normal bundle to $M$ in $N$ to $\partial_1 M \setminus \partial_1 N$, but then you have to take a direct sum with $[0,\infty)$. Here $\partial_1 M$ means the co-dimension one boundary stratum of $M$. Similarly $\partial_2 M$ means the co-dimension two strata, i.e. the traditional `corners', for example $\partial_2 [0,\infty)^2 = \{(0,0)\}$.

The co-dimension 2 stratum of the regular neighbourhood consists of the normal bundle to $M$ in $N$ restricted to $\partial_2 M$, but then you have to take a direct sum with $[0,\infty)^k$ depending on which stratum of $M$ you are in. $k=2$ for the interior stratum, $k=1$ for the co-dimension one stratum and $k=0$ for the co-dimension two stratum.

etc.

These spaces you glue together in the natural way, and you have a corresponding embedding from this total space to the ambient manifold $M$.

I could revise this to be more precise but let me know if that makes sense. It corresponds to your images, but my response deals with some cases you did not include in your images.