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In both examples you are considering, the anticanonical model is a singular del Pezzo surface.

In fact, let $X$ be the blow-up of $\mathbb{P}^2$ at three points lying on a line $L$. By Bezout's theorem, the birational map associated with the linear system of cubics through the three points contracts $L$. Since $L^2=1$, the blow-up of $L$ at three points gives a $(-2)$-curve. Therefore, the anticanonical model of $X$ is a Del Pezzo of degree $6$ in $\mathbb{P}^6$ with an ordinary double point (i.e., a node).

Analogously, let $Y$ be the blow-up of $\mathbb{P}^2$ at six points lying on a conic $C$. By Bezout's theorem, the birational map associated with the linear system of cubics through the six points contracts $C$. Since $C^2=4$ and we are blowing-up six points over $C$, we obtain again a $(-2)$-curve. Therefore, the anticanonical model of $Y$ is a Del Pezzo surface of degree $3$ in $\mathbb{P}^3$ with an ordinary double point, i.e. a cubic surface with a node.

Of course, if you blow-up more than $8$ points then the result is not a Del Pezzo anymore. For instance, the blow-up of $\mathbb{P}^2$ at nine points which are the base locus of a pencil of cubics is an elliptic fibration $X \to \mathbb{P}^1$ with nine sections; in general, such fibration has exactly $12$ nodal fibres, corresponding to the singular elements of the pencil.

When the number of points increases the situation becomes more and more complicated, and I guess that a satisfatory description is out of reach.
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In the two cases both examples you are asking forconsidering, the anticanonical models are model is a singular del Pezzo surfacessurface.

In fact, let $X$ be the blow-up of $\mathbb{P}^2$ at three points lying on a line $L$. By Bezout's theorem, the birational map associated with the linear system of cubics through the three points contracts $L$. Since $L^2=1$, the blow-up of $L$ at three points gives a $(-2)$-curve. Therefore, the anticanonical model of $X$ is a Del Pezzo of degree $6$ in $\mathbb{P}^6$ with an ordinary double point (i.e., a node).

Analogously, let $Y$ be the blow-up of $\mathbb{P}^2$ at six points lying on a conic $C$. By Bezout's theorem, the birational map associated with the linear system of cubics through the six points contracts $C$. Since $C^2=4$ and we are blowing-up six points over $C$, we obtain again a $(-2)$-curve. Therefore, the anticanonical model of $Y$ is a Del Pezzo surface of degree $3$ in $\mathbb{P}^3$ with an ordinary double point, i.e. a cubic surface with a node.
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In the two cases you are asking for, the anticanonical models are singular del Pezzo surfaces.

In fact, let $X$ be the blow-up of $\mathbb{P}^2$ at three points lying on a line $L$. By Bezout's theorem, the birational map associated with the linear system of cubics through the three points contracts $L$. Since $L^2=1$, the blow-up of $L$ at three points gives a $(-2)$-curve. Therefore, the anticanonical model of $X$ is a Del Pezzo of degree $6$ in $\mathbb{P}^6$ with an ordinary double point (i.e., a node).

Analogously, let $Y$ be the blow-up of $\mathbb{P}^2$ at six points lying on a conic $C$. By Bezout's theorem, the birational map associated with the linear system of cubics through the six points contracts $C$. Since $C^2=4$ and we are blowing-up six points over $C$, we obtain again a $(-2)$-curve. Therefore, the anticanonical model of $Y$ is a Del Pezzo surface of degree $3$ in $\mathbb{P}^3$ with an ordinary double point, i.e. a cubic surface with a node.