The sources I'm looking at are giving me conflicting information. One paper gives the presentation $$\langle x,y|xyx=yxy, x^2y=yx^2\rangle,$$
while another paper asserts that Example 12 from Fox's A Quick Trip through Knot Theory is in fact the two-twist spun trefoil, which has the presentation $$\langle x,y|xy^2=yx, y^2x=xy\rangle=\langle x,y|y^3=1, xyx^{-1}=y^{-1}\rangle.$$
Finally, a third paper gives the presentation as $$\langle x,y|xyx=yxy, yxy^{-1}=xyx^{-1}\rangle$$ citing a paper by Zeeman that appears to only talk about the 5-twist spun trefoil.
Through a little bit of work, I can see that the first and third groups are the same, but the second one seems distinct from them both. Also, if I use these presentations to construct the first Alexander ideal, I get different ideals, and as far as I know, the 'right' ideal should be $\langle 3, t+1\rangle$. Is there a definitive source as to which of these are right, or where my confusion might be coming from?
All of these presentations are right and present isomorphic groups.
You will get the second presentation in the form $\left<x,t | t^3=1, xtx^{-1}=t^{-1}\right>$ from the first presentation by substitution $t=yx^{-1}$.