join in vs join

[grammarguy]

I am really confused about the difference between "join" and "join in". I am going to make up two pairs of sentences below.

(1a) I would like to join you for lunch.
(1b) I would like to join in the lunch next week.

(2a) I would like to join my friends for your birthday party.
(2b) May I join in your birthday party next week?
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This is how I interpret them.

You say the a's when you want to go with someone to a lunch or a party.

Suppose that your colleague is asking who would like to have lunch with him next week. You would say (1b).

Suppose that your neighbor is telling you that he has invited some people to his birthday party next week. You would say (2b) if you asked for his invitation.
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May I ask two questions?

(A) Are my interpretations correct?
(B) In the scenarios for the b's, is it idiomatic to say "join in + an object" as in "join in the lunch and join in the party"?

Please answer my questions. I am very confused about their correct usage. Thank you very much for your help.

[Alan]

(Not really a grammar issue but...) Essentially, 'join' is for groups, and 'join in' is for activities/events.