Relation between Lens spaces and Hopf fibrations

Is a Lens space $$L(p,q)$$, for some particular values of $$p$$ and/or $$q$$, homeomorphic to an $$S^3$$ arising as a Hopf fibration?

For instance, from 1903.04942:

[...] the $$S^1$$ bundle over $$S^2$$ with Hopf invariant $$N$$ -- i.e. the Lens space $$L(N,1)$$.

This seems like a realistic claim to me, because it would mean, for example, that the circle bundle over $$\mathbb{E}^3\setminus0$$ whose total space is the single-centre $$(N=1)$$ Euclidean Taub-NUT space would restrict to an $$L(1,1)$$ topology on a constant radius surface in the $$\mathbb{E}^3$$, and indeed it is known that this restriction is an $$S^3$$ realised as a Hopf fibration (see e.g.hep-th/0208108) (and the topology of the $$S^1$$ fibration over $$S^2$$ is $$S^3$$, and $$S^3$$ is homeomorphic to $$L(1,q)$$).

What I'm also confused about is that $$L(1,p)\simeq S^3$$. From the claim above, it would seem to me that, to have $$S^3$$ as a Hopf fibration with invariant $$N$$, I have to set $$p=1$$, and then the first parameter of the Lens somehow encodes the integer classifying the Hopf fibration?? Thanks.

I think that what creates some confusion here is the different notations that authors use throughout the literature. I believe that the notation used in the first claim you mention, $$L(p,q)$$ is defined as a $$\textit{quotient of}$$ $$S^3$$ (with $$S^3: |z_0|^2 + |z_1|^2 =1$$) obtained via the equivalence relation $$(z_0,z_1) \sim (e^{i2\pi /p} z_0, e^{i 2 \pi q / p} z_1)$$. This is a very common definition (for instance, it is the one you would find on Wikipedia, where you can also find some references). With this definition, then the claim that $$L(p,1)$$ is the Hopf bundle with Hopf invariant $$p$$ is correct.
Unfortunately, there is another definition which is sometimes used, where the space $$L(t,s)$$ is defined as the quotient space of $$S^{2t+1}$$ obtained via $$(z_0,...,z_t) \sim (e^{i2\pi /s} z_0,..., e^{i 2 \pi / s} z_t)$$. This definition is given for instance in the famous book by Bott and Tu. I am pointing this out just to make you aware of this annoying two-notations issue in the literature.
Regarding the second claim, i.e. $$L(1,p) \simeq S^3$$, could you be a bit more specific about the context in which you found it?