How about this: two objects are touching when the interaction forces between the objects are of similar magnitude to the interatomic forces within the objects. Very generally speaking, this will happen when the distance between the objects is about the same as the distance between atoms within the objects.

To be somewhat pedantic: "touching" is a word. The extent to which it is or is not an "exact science" depends on what is meant by "touching", and by "exact science". The definition I gave of "touching" could be made arbitrarily more precise by specifying more precisely the interaction forces under consideration, and the level of precision to which one should compare their magnitudes. What you mean by "exact science" is something I cannot guess at with any useful level of precision. Whether or not any of this has any bearing on the original topic of discussion is another matter.

I mean, the actual interaction force that defines "touching" is open for debate. Touching is an abstract concept, removed from actual physics.

That was sort of my point. If we use a definition of "touching" that is relevant to the atomic scale, whatever that definition is, we can reasonably talk about what "touching" means at an atomic scale. If we use a colloquial definition of "touching" which is meaningful for macroscopic objects but not at the atomic scale, it doesn't make sense to talk about "touching" at the atomic scale.

When weak interaction starts happening.

When does the weak interaction stop happening?

High energies, about three orders of magnitude higher than the LHC produces. (If you squint.)

Do this for 'touching fire' to clarify the actual issue.

There isn't an exact moment, any more than there is an exact height that is "tall". And yet, tall people exist.

The tallest person in the world will say that only normal and short people exist.

I'd bet very good money that they wouldn't say that, no