When selecting a product to use for digestive bitters, does it matter which herbs the product contains or are we just concerned with the fact that it has a bitter taste?

It will depend if you follow the directions on the bottle or not.  Most bitters recommend a full teaspoon.  Since most bitters contain stimulant laxatives and/or berberine herbs like goldenseal and Oregon grape root a full teaspoon presents problems.  Since a person only has to taste the bitterness for bitters to work I recommend a half a dropper full, which is enough to taste the bitterness without developing problems from the stimulant laxatives or berberine.

For example, is the digestive system stimulated to a different extent when using artichoke extract as opposed to swedish bitters, etc.

No, both stimulate the vagus nerve through the bitter receptors on the tongue and therefore work the same.

Do you think there is a product that stands out above the rest?

Chinese Bitters only have gentian root for a bitter.  No stimulant laxatives or berberine.  I have not seen these in health food stores though.  I think you can only get them on line.  If you want a bitter from a health food store so you can get them right away I recommend Grape Bitters by Planetary Formulas or Gallexier Bitters by Flora.  The other option is to just get some bitter herbal powder such as gentian root, put a pinch of the powder on your tongue.  Then suck on the powder for a few seconds and swallow.  This is a good choice for people also wanting to avoid the alcohol or sugar in some bitters formulas.