"the kids" X and J always works well. Or just don't use any term and say this is "X" and this is "J" There is really no reason to go into detail with casual introductions. Your close family and friends already know the relationship and won't need an explanation. I have always found that is leads to a bunch of ridiculous questions when you label relationships beyond the adult relationships. 

You wouldn't typically invite BM's family to a party for your child. Your SO can always have a separate birthday party. I think you are overthinking this. 

I think you are totally wrong here. While I understand you not wanting to claim any ownership of your BF's other son, the rest of what you are saying about trying to keep the kids apart and not family is not ok. Sure, introduce the kids as "This is our son"DS Name" and this is my BF's/Fiance/ DH's son "SS Name"

Like it or not, your BF's other son is your child's brother. My eyes just can't roll dramatic enough when you come back with- "ONLY 1/2!"- but trust me, my eyes are rolling. I ONLY have 1/2 siblings. They are my sisters and brothers. The end. I only grew up in a household with 2 of them. The other 3 lived with other parents. IT DOES NOT MAKE THEM LESS. My older sister (we share a dad)- guess who is helping me my 1st day post op hysterectomy next week?? Yup, one of those evil 1/2 siblings. OH! Guess who is taking SS to and from his vocational school while I can't drive??? Another one of those dreaded 1/2 sisters! Who is a chef and has loaded my freezer with meals for 2 weeks? Ugh... one of those nasty 1/2 brothers! LMAO

 My parents always encouraged my relationships with MY FAMILY. Even though those kids aren't related to my mom and "stepdad" (who adopted me when I was 7 and is the only dad I will ever acknowledge)- they are my sister and brothers. We are here for each other. We love each other and we have healthy relationships. My niece calls my dad PAPA- even though there is literally NOTHING tying them together. My nieces kids- run and hug my parents when they see them. My older 1/2 sister and her husband have spent many Thanksgivings and Christmases with my side- no one skips a beat. They are my family- therefore they are all family. 

Open you heart instead of being a jerk to a kid that has no control over his parents. 

You are making this way too complicated.. it's not all that big of a deal.

If your BF's kid has a birthday party at his mom's... YOU likely won't be invited.. neither will your boyfriend.  Likewise, you are not obligated to have her at YOUR house if your BF throws a party for his son. 

Whether you allow YOUR baby to go to BM's house for a birthday party?... it will just depend and will be quite dependent on the relationship the kids have.

And.. about that relationship.  STOP being sellfish.  Your son should have the opportunity and the blessing of having a sibling relationship.  You and your BF should absolutely encourage that to the extent possible.  There will be far more benefits to your chilld having a healthy relationship with a sibling.. that will be there long after you and your BF are dead and gone. 

That means.. NO you can't exclude your BF's son.. you can't keep him away from things like Bday parties that siblings normally do.  That means that he will be welcome on 'family' vacations to the extent possible.. he may not be able to be there for everything.. but you don't intentionally plan everything when he CAN'T be there.

Do YOU have to love your ss? no

Do YOU have to financially support your SS? NO

Do YOU have to raise/discipline your SS? again.. NO.

But he should be welcome in his father's home...and I am assuming that is where you live.

I would essentially look on how is the BF’s boy and what is his education. If all of it is acceptable, I would give it a shot. 

In my case the DH’s boy is given a very particular education and has a very particular behavior. And on a few occasions I had my DD close to him, I have noticed that she tends to copy. As an example, our house is very quiet and we can run and play and make noise outside of the house. In house we are all calm, I don’t know why but it is the case and I like it. The boy is constantly screaming, everywhere. ILs are screaming too (because it is impossible to talk without screaming when the boy is around. My daughter started to scream in their presence too :( 

also, he likes to through himself on the floor and hit his head against it if he doesn’t get what he wants. It could be really scary, a kid who does that. And the most important , my ILs would give him what he wants, to stop this behavior. Well my daughter saw it and also tried. It didn’t work with me, the ILs wanted to allow her to have what she wanted, I was pissed off and didn’t let them. 

Long story short, I see that the boy has a completely different education than what I try to give to my DD. I know I cannot have much impact on the boy but at least I can protect my daughter and I do so. 

And in my own childhood my father’s boy from previous relationship was also a bad example for me. Parents used to  force me into half-sibling relationship and it didn’t work well. Fortunately, they woke up when the boy tried to make me smoke weed and showed me porn. He was finally banned from our house and father was seeing  him at his parents’ house. I do not call them GPs because I was banned from their house as they were hoping that my father would get back with the previous lady to be around the first boy. 

It was a total mess for all of us, to have this forced relationship, to be honest. I would never do that to my daughter. If one day she is interested in this boy and wants to have some interactions, she will have them. But I myself will not force anything in that direction