This question is about two definitions of enriched monoidal categories I have:

Let $\mathcal{V}$ be a symmetric monoidal closed category.

The first definition: a $\mathcal{V}$-enriched category $\mathcal{C}$ is a pseudomonoid object in the Day-convolution monoidal category $(\mathcal{V}\text{-}\mathbf{Cat}, \otimes_{\mathrm{Day}}, y(1))$. This is a straightforward generalization of the definition of monoidal categories but everything has to be worked externally and I don't really feel like to work in this definition (for example, I am pretty sure that a kind of coherence theorem holds but I don't know if I will be able to write down the proof).

The second definition can be found here (Bar constructions for topological operads and the Goodwillie derivatives of the identity, Definition 1.10): https://projecteuclid.org/download/pdf_1/euclid.gt/1513799607

In this paper, an enriched monoidal category $(\mathcal{C}, \overline{\wedge}, S)$ over $(\mathcal{V}, \wedge, I)$ is a $\mathcal{V}$-enriched category $\mathcal{C}$ which is tensored (denoted by $\otimes$) and cotensored, together with a monoidal structure on the underlying category $\mathcal{C}_0$ and the distribution natural transformation $d: (X\wedge Y)\otimes (C \overline{\wedge} D)\to (X\otimes C)\overline{\wedge}(Y\otimes D)$ satisfying certain associativity and unitality condition, which is a natural requirement for bar constructions.

So my question is: What is the relation between these two? It will be pleasing to be able to replace the first definition by the second one in other situations but I could not see any immediate implication from one to another or confluence under some conditions.