What is the difference between these pairings classifications?

I know the basic definitions of bilinear groups. For example, there is a bilinear pairing that uses elliptic curves and has the following properties: For $$G_1$$, and $$G_2$$ are cyclic groups of prime order $$p$$ with $$P_1$$ is the generator of G_1 and $$P_2$$ is the generator of G_2. And there is $$e : G_1 × G_1 → G_T$$, meaning that (1) For all $$P ∈ G_1$$, $$Q ∈ G_2$$ it holds that $$e(aP_1,bQ )=e(P,Q)^{ab}$$ (2) $$e(P_1,P_2)≠1$$;

But I have just read this in a paper https://infoscience.epfl.ch/record/128718/files/CCS08.pdf, that is based on BLS signature https://www.iacr.org/archive/asiacrypt2001/22480516.pdf. And the paper stated that"

Let PG be a pairing group generator that on input $$1^k$$ outputs descriptions of multiplicative groups $$G_1$$ and $$G_T$$ of prime order $$p$$ where $$|p| =k$$. Let and let $$g \in G_1^{*}$$ and $$G_{1}^{*}=G_{1} \backslash\{1\}$$ .The generated groups are such that there exists an admissible bilinear map $$e : G_1 × G_1 → G_T$$, meaning that (1) for all $$a, b ∈ Z_p$$ it holds that $$e(g^a,g^b )=e(g,g)^{ab}$$ (2) $$e(g,g)≠1$$; and (3) the bilinear map is efficiently n.

What is the difference between the first pairing group that uses elliptic curves and the second one? How could I name the second pairing groups? Could anyone suggest examples for the two examples?