A creature that touches a Contact Poison with exposed skin suffers its effects.
Does a successful melee attack satisfy this “touching” requirement?

RAW: No. Specific Rules trump implied rules.

P257 of Dungeon Master Guide lists the poison types and some samples of those. There is a type for Contact, among others is a type for Injury.
Injury states

A creature that takes slashing or piercing damage from a weapon or
piece of ammunition coated with injury poison is exposed to its
effects

Explaining that it can be used as you've mentioned, with melee weapons. So that's a specific rule that shows how these poisons can be delivered.

Common Sense Ruling: Yes

Same page as noted above, the listing for Contact type poison states:

A creature that touches contact poison with exposed skin suffers its
effects

Note that Injury type lists slashing or piercing damage type... it doesn't mention bludgeoning. That is probably where I would interpret that Contact poison type could be used.

If you look at the sample types of poison on the following page in the DMG, it has extremely similar samples of poison for Injury and Contact. So technically if you wanted to simplify this for ease of use, a DM could rule that they're the same "type" of poisons... meant to be used in combat.

Yes, the poison can be used with anything that can touch skin

The description for contact poisons say:

A creature that touches contact poison with exposed skin suffers its effects

That is all the guidance the rules offer for delivering the poison.

Can a weapon contact bare skin? Of course it can. So can a glove or letter or an apple. Therefore all these things can be used to deliver contact poison.

There are obviously a near infinite number of ways that you could go about contacting bare skin and the rules wisely do not try to enumerate them. One way to contact skin is to forcefully contact it with something else. This something else could absolutely be a weapon and the method of contact could absolutely be an attack.

DM will have to come up with a method of adjudication

The only tricky part about adjudicating this poison as a whole is that it is entirely up to the DM as to how to determine if anything successfully contacts skin. There are no hard rules as to how to deliver the poison and thus the DM will have to fill in the gaps with other rules. This covers all applications of the poison including trying to deliver it with a weapon.

When you make an attack with a weapon coated in contact poison, your DM will have to find a way to make a ruling on if that attack manages to contact bare skin. This will likely be harder on some enemies than others. A barbarian in only loincloth? Barely a challenge. A paladin decked out in full plate? Much harder.