As long as you treat these as two separate events what you are planning is just fine. You aren’t having two wedding receptions. You are having the wedding (i.e. the ceremony) followed by a reception (i.e. cake and punch, mingling, conversation and fellowship) — which is all very traditional. That all goes on one invitation card that goes out to everyone. We great-relatives are used to this: it is how receptions were done routinely up until the last twenty years or so.
Later in the day, you are having a “dinner party”. You will doubtless hold many dinner parties throughout your married life, all of which will have different guest-lists from the guest-list at your wedding. This is just the first of them. You must have it at a different location and after a substantial time gap, so that you aren’t rushing away from your wedding guests and aren’t setting up tables somewhere in view in a manner that might make them feel like they are being left out. You have a separate card for this separate event, which you may include in the same envelope as the wedding invitation if you wish. You may refer to the second event just as “Dinner” or if you wish you may refer to it by its traditional name as a “Wedding Breakfast” (and risk all guests who don’t know the actual meaning of “breakfast”, expecting to be served bacon and eggs with oatmeal).
If you are being formal, the second card can read:
” Mr John Host and Ms Jane Hostess
“request the pleasure of your company
” to Dinner
” at six o’clock on <date>
” at the Waldorf Hotel”
If you word it this way, you can send it in a separate envelope, which further reduces the risk of it’s being seen as a continuation of the previous event. If you include it in the wedding invitation envelope, however, you may leave out the first two lines and have the card simply printed:
” six o’clock
“at the Waldorf Hotel”
If your friends DIDN’T learn (back in Kindergarten) not to flaunt invitations in front of people who may not be invited, you might want to drop them a hint by saying “Private Dinner”, or “Private Wedding Breakfast”, or even “Intimate Dinner”. There are problems with those wordings, and I wouldn’t consider using them myself, but you have to play to the crowd that you are dealing with.
And meanwhile we great-relatives may very well choose to go out to dinner and dancing with one another since we’re in town. No offense to you the bride, but we are interested in catching up with one another too, and are entitled to plan our own “separate events”. Any great-relatives who get their noses out of joint by hearing there’s an event in town they aren’t invited to need to grow up and take some sophisticated responsibility for their own social lives.