Dining does not only refer to dinner: a hotel will usually serve lunch in the dining room, for example. But if you say you plan to dine out, most people will take that to mean go out for an evening meal.
But except in set phrases like dine out, dine out on, dining room, dine with the devil, he wined and dined her, it is a very formal word, and you would not use it in an informal conversation about joining someone for lunch, and if you say come and dine with me to invite someone to dinner, you will sound as though you have been reading too many historical novels.
And... not join with you for the lunch, just join you for lunch.