The value of t in r1 is not the same as the value of t in r2 when the curves intersect. Here's one way to find the intersections:
In[1]:= r1[t_] = {2 Cos[t], 2 Sin[t]};
In[2]:= r2[t_] = {3 Cos[t], Sin[t] - 1};
In[17]:= r =
Reduce[{r1[t1] == r2[t2], 0 <= t1 <= 2 \[Pi],
0 <= t2 <= 2 \[Pi]}, {t1, t2}]
Out[17]= (t1 == (3 \[Pi])/2 &&
t2 == (3 \[Pi])/2) || (t1 == 2 \[Pi] - 2 ArcTan[8 - 3 Sqrt[7]] &&
t2 == 2 ArcTan[(-53 + 20 Sqrt[7])/(3 (-8 + 3 Sqrt[7]))]) || (t1 ==
2 \[Pi] - 2 ArcTan[8 + 3 Sqrt[7]] &&
t2 == 2 ArcTan[(53 + 20 Sqrt[7])/(3 (8 + 3 Sqrt[7]))])
In[23]:= r1[t1] /. {ToRules[r]}
Out[23]= {{0, -2}, {2 Cos[2 ArcTan[8 - 3 Sqrt[7]]], -2 Sin[
2 ArcTan[8 - 3 Sqrt[7]]]}, {2 Cos[
2 ArcTan[8 + 3 Sqrt[7]]], -2 Sin[2 ArcTan[8 + 3 Sqrt[7]]]}}
In[24]:= N[%]
Out[24]= {{0., -2.}, {1.98431, -0.25}, {-1.98431, -0.25}}