You should ideally choose the test specification based on subject-matter knowledge before looking at the results. Or you could test which specification is the right one, but that is pretty tricky and may lead to finding cointegration more often than it should be (as you allow the test specification to adapt to the behaviour of the series too well).
One may assume a stock price index to follow a random walk with or without a drift. A drift is justified from the financial theory point of view as a compensation for undertaking risk (compare the "risky" investment in stocks to a "safe" investment in U.S. government bonds).
- If you assume upfront that the series does not have a linear trend (i.e. a drift), you should just neglect the test result for that case.
- If you assume the series has a linear trend (i.e. a drift), you have to deal with the test result for that case.
How come the unit root tests suggest integratedness while the cointegration test suggests no integratedness? This conflict is of course confusing and could not hold in population. Nevertheless, it may appear in finite samples where the unit root test is not powerful enough to reject a unit root while the cointegration test has enough power to reject lower orders of cointegration.
You could either try alternative tests looking for which case receives more support or just assume what you believe to be right and proceed.