It seems to me that there are two conventions of defining (differential) $k$-forms, namely depending on whether the factor $1/k!$ is included in front of the alternating sum, so e.g. for $2$-forms one can either have $$ dx \wedge dy = \frac{1}{2!} (dx \otimes dy - dy \otimes dx) \quad\text{or}\quad dx\wedge dy = dx \otimes dy - dy \otimes dx. $$ Depending on which of the two one chooses, the formula for wedge products between $k$-forms also changes of course.
I am more interested in the implications this choice has once one starts introducing (semi-Riemannian) inner products. A metric $g$ on the manifold induces an inner product on $1$-forms - in coordinates $\langle dx^i, dx^j \rangle = g^{ij}$. This can be extended to higher order tensors simply by multiplying the products of the components, i.e. $\langle dx^{i_1} \otimes \cdots \otimes dx^{i_k}, dx^{j_1} \otimes \cdots \otimes dx^{j_k} \rangle = g^{i_1 j_1} \cdots g^{i_kj_k}.$ Now, depending on which one of the two conventions for defining forms one chooses, we either have $$ \langle dx^{i_1} \wedge \cdots \wedge dx^{i_k}, dx^{j_1} \wedge \cdots \wedge dx^{j_k} \rangle = \frac{1}{k!} \det \langle dx^{i_a}, dx^{i_b}\rangle \quad \text{ or } \quad k! \det \langle dx^{i_a}, dx^{i_b}\rangle. $$ But I have often seen this also defined as $$ \langle dx^{i_1} \wedge \cdots \wedge dx^{i_k}, dx^{j_1} \wedge \cdots \wedge dx^{j_k} \rangle = \det \langle dx^{i_a}, dx^{i_b}\rangle, $$ which if I understand correctly, one obtains if one instead uses $\langle dx^{i_1} \otimes \cdots \otimes dx^{i_k}, dx^{j_1} \otimes \cdots \otimes dx^{j_k} \rangle = \frac{1}{k!} g^{i_1 j_1} \cdots g^{i_kj_k}$ and the second convention for defining forms stated above.
I am a bit confused about the importance of these choices. For example, it seems to me that these then also have different definitions of the Hodge star operator and consequently also the volume form. Does this affect integration on manifolds? E.g. is the volume of the unit $2$-sphere equal to $4\pi$ with respect to all these conventions? Is there anything else they affect?