Private ceremony/big reception?

The whole point of the day is marriage, the wedding, not the reception. It's like inviting people to a birthday party after you sang and blew out the candles. Please invite everyone to both, it is rude not to.

It’s completely up to you! I’ve been to a reception only wedding because they got married in a court house, I didn’t find it rude. I personally would want to go to both, but it all depends on you and your guests. Out of town guess should most definitely be at both, because a long drive/flight wouldn’t be worth it in my opinion just for one

If your private ceremony is the same city, on the same day and right before your reception I think it's a little umconventional to do what your proposing. I personally would be a bit bummed if someone invited me to the reception only. If someone asked I would probably say "oh I understand. That's fine, we'll see you at the reception!," but would probably be hurt I didn't make the exclusive "in club".
I'm not sure I quite understand why you wouldn't want everyone at the ceremony with you. Or how a "private" ceremony is signifacantly affected by inviting everyone. You could block off the first few rows for your VIPs. Is there a certain venue you have in mind that only seats a few people?
The fact you posted the question, even after asking some, makes me think you already know it might be a bit rude.

It's fine if you invited ONLY immediate family (parents, siblings with their SO and kids, and grandparents) to the ceremony. But once you invite more.than that to the ceremony, it becomes rude to exclude the rest of your guests.

A lot of people are now going this route for several reasons. I guarantee someone will find it rude/hurtful on your wedding day. That being said, most of your friends/family should know you well enough that they'll at least understand. Some will also just be happy you "stuck to you."

My only concern is when you add in "close friends." i suggest making those close friends a part of your bridal party. Otherwise, that WILL feel like a slap in the face to a lot more people.

I think it's rude and hurtful not to include people to the ceremony but want them to come to the reception. I think you give your guest the opportunity to be witnesses that day by putting the ceremony and reception time on the invite and letting them chose what they want to attend is probably the better option.

It’s perfectly fine to have an intimate ceremony with only immediate family and then invite everyone else to the reception. The only exception would be a small bridal party including friends. Otherwise, you can’t invite a few friends to the ceremony and not others. It’s all or nothing.

As long as they arent in the same place it should be ok. Just make it clear on the invites that you are inviting them to the reception only. Something like we invite you to the "celebration of our marriage". Its what people who have small destination weddings and larger receptions back home do.