Vinegar choice for pickling
While you could certainly use plain white vinegar, there's not all that much difference between that and cider vinegar if "vinegar" is the taste you're objecting to.

If you are canning the pickles, changing the recipe could kill you. The vinegar helps to produce an acid environment where botulism does not grow.

So, if it turns out you don't like pickles in vinegar, stick to "refrigerator pickles" if you want to reduce the vinegar further. That is, keep them in the refrigerator the entire time.

You might also want to look into lactic (naturally fermented) pickles where an appropriate salt level keeps things safe until the lactic acid bacteria make enough acid to further keep things safe. Those are refrigerated when complete, typically.

Waiting longer (rather than throwing away) may help the ones you have - it takes some time for the exchange between the inside of the cucumber and the brine to be complete. Or, a refrigerated dip in fresh water for overnight or a few days might help them suit your taste.