The motive $H_2(\overline{M}_{0,5}\setminus A, B\setminus A\cap B)$ seems to be the right one. The relative singular cohomology is 2-dimensional as one would expect from an extension in $Ext^1(\mathbb{Q}(0),\mathbb{Q}(n))$. The computation can be found as Theorem 5.3 in

However, it should be pointed out that just looking at the dimension of the Betti realization is not the right thing anyway. For example, there are also motives of the form $H_2(\overline{M}_{0,5}\setminus A, B\setminus A\cap B)$ (for other divisors $A$ and $B$) which realize dilogarithms $Li_2(x)$ for $x\neq 0,1$. The corresponding relative singular cohomology groups are 3-dimensional, again by Theorem 5.3 of the paper of Wang mentioned above. This would be an example where the Betti realization of the motive is too big for an extension.

The extensions in $Ext^1(\mathbb{Q}(0),\mathbb{Q}(n))$ are realized by *framed motives*, so the point in the paper of Goncharov and Manin is not just the construction of a motive but also of an appropriate framing. The framing of $M=H^n(\overline{M}_{0,n+3}\setminus A, B\setminus A\cap B)$ consists of elements $[\Omega_A]\in Gr^W_{2n}H^n(\overline{M}_{0,n+3}\setminus A)$ (given more concretely by a differential form) and $[\Delta_B]\in (Gr^W_0 H^n(\overline{M}_{0,n+3},B))^\vee$ (given as a relative cycle). The integral of the differential form over the cycle gives the zeta-value, and this is the sense in which the framed motive realizes the zeta-value. If the dimension of cohomology is bigger than 2, framing the motive corresponds to picking a $(2\times 2)$-submatrix of the period matrix which contains $\zeta(n)$.
The framing described above corresponds to supplying maps $\mathbb{Q}(-n)\to M$ and $M\to\mathbb{Q}(0)$ which in the case $\zeta(2)$ arise from the relative motive exact sequence
$$
H^1(B\setminus A\cap B)\to H^2(\overline{M}_{0,5}\setminus A,B\setminus A\cap B)\to H^2(\overline{M}_{0,5}\setminus A)
$$

I think the short answer to the question is that framed motives can be used to describe extensions in $\operatorname{Ext}^1_{\operatorname{MT}(k)}(\mathbb{Q}(0),\mathbb{Q}(n))$, but for this is not necessary for the underlying motive of the framed motive to have the ``right dimension''. In general, only a subquotient of it will give the extension (provided the vanishing of a coproduct). The essential idea (going back to Beilinson-MacPherson-Schechtman's notes on motivic cohomology, it seems) is that there is an equivalence relation on framed motives such that each equivalence class contains a minimal representative corresponding to an actual extension (and hence equivalence classes of framed motives describe Ext-groups of motives). General discussion of framed motives can be found in

A.A. Beilinson, A.B. Goncharov, V.V. Schechtman and A.N. Varchenko: Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane. Grothendieck Festschrift I, 1990, pp. 135-172.

A.B. Goncharov. Periods and mixed motives. http://arxiv.org/abs/math/0202154

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