A Lawvere theory on a category $C$ is a functor $L:\aleph_0^\text{op}\to C$ preserving finite products which is a bijection on objects, where $\aleph_0^\text{op}$ is the skeletal category of finite sets.
What is an example of a category $C$ and two different lawvere theory structures on $C$?
I can imagine that simple examples can be obtained by freely adjoining some arrows to the category $\aleph_0^\text{op}$ to get ${\aleph_0'}^\text{op}$ and look for finite product preserving functors $\aleph_0^\text{op}\to{\aleph_0'}^\text{op}$. However I have little to no intuition for what this means, which is why I am asking for an "interesting" example of this.
However, I am also suspicious of my claim because my understanding is that a model of a Lawvere theory $L:\aleph_0^\text{op}\to C$ are finite product preserving functors $C\to \mathcal Sets$, which do not depend on the choice of $L$. This would mean that two different Lawvere theories have the same models, which feels wrong to me.
My conclusion is that either my claim is wrong, and the structure of a Lawvere theory on $C$ is uniquely determined, or my understanding of Lawvere theories is flawed.
The functor $L$ in your definition is often assumed to be the identity on objects. In that case it's definitely uniquely determined, because the only morphisms of $ℵ_0^{\mathrm{op}}$ are tuples of projections, which are preserved by $L$ since it preserves products. You can also ask for $L$ to be essentially surjective instead, in which case it won't be uniquely determined in general, but it will be unique up to isomorphism, which isn't substantially different.
Calling any functor $L : ℵ_0^{\mathrm{op}} → C$ which preserves finite products a Lawvere theory doesn't make much sense.
In any case, $L$ is just a matter of convenience, all the required data is already present in $C$. You can just as well say that a Lawvere theory is a category $C$ with distinct objects $\{T_0, T_1, T_2, ...\}$, such that $T_n = T_1^n$. It's basically the same thing.
While it doesn't make sense to speak about two different Lawvere theories "on" $C$, it makes sense to ask if two different Lawvere theories $C$ and $D$ can have equivalent categories of models. The answer is most definitely yes, and the question an important one. Such equivalence is called Morita Equivalence of theories, because it directly generalizes the Morita equivalence of rings. For example, Lawvere theores for right $ℝ$-modules and right $M_nℝ$-modules are certainly different, but the categories of models are equivalent.