Phrasal verbs join in; work out

Sentences:

-s1- I have been joining in at the gym (wrong)
-s2- I have been working out at the gym (right)

Issues:

-i1- I am not sure why s1 is wrong. I think may be because of the “at” particle (In this case “I have been joining in the gym” would be correct). Please could you explain or confirm?

-i2- join in means to participate. Is this right?
-i3- work out means to do exercise, right?

Thanks in advance !!

.
1-- ‘Join in’ is for a group activity, but working out is an individual effort merely performed in the same room.

2-- Yes, but it does not mean ‘become a member’-- that is ‘join’ only.

3-- Yes, exercises on machines or with free weights, mostly.
.

I entirely agree with you, Mister Micawber