Using the word 'back' in a headline [duplicate]

Solution 1:

I'm not sure what part of speech 'back' is in these examples.

In this context, the word back is an enclitic. It changes the verb to welcome into the verb to welcome back. Many prepositions are used this way in English, to add some directional modification to a verb. (Other examples: to bring in, to post up, to grind down, and so forth.)

I see little difference between your options 1 and 2. There is no significant difference between the usages.

I will venture to share my take on the two options:

Option 1 sounds like "We" have been welcoming back various entities, and now "We" are welcoming families back. (We previously welcomed students back, and welcomed teachers back, and welcomed administrators back.) To my subjective ear, it sounds like "We" are the ones who are choosing whom to welcome back, as "We" please. "We" are the agents in this usage.

Option 2 sounds like "We" are welcoming whomever comes back, as they are pleased to choose to return. To my subjective ear, this usage sounds like the ones who are returning are choosing when to come back, and "We" are welcoming all, as they come. "They" are the agents in this usage.