It is not necessarily rude per se.
It is ALL about how you do it that will or will not make it appear rude.
There is nothing wrong with having an Adults Only Reception… and you can write that on your Recption Cards / Invites as follows “Adult Reception to follow”
Gernerally speaking the Wedding Ceremony itself is OPEN to all… (because “technically” a house of worship is supposed to be a Welcoming Place for all). So there is a good chance that you’d have kiddies there no matter what… be they from your Relatives or Friends attending.
And then no Kiddies at the Adult Reception (except for those in the Wedding Party for the “meal” portion). You would have to have a chat with those Parents as well… letting them know that as of say 8 PM it would be the Adults Only (as per the wording on your Invites / Reception Cards… “Adult Reception Only”)
Of course, you could have NO KIDS whatsoever at the entire Wedding and Reception… the first step to doing that is having the Invitation Envelopes clearly marked for WHO is invited (Mr & Mrs Robert Jones)… and then follow it up with the RSVP Cards, that you have pre-marked as
___ of 2 Guests WILL BE / WILL NOT BE Attending (please circle)
And of course you’d have to have a sit-down with the Parents of those in the Bridal Party explainging that there will be NO KIDS WHATSOEVER at the Reception… and that you’ve made arrangements for the tykes to be babysat / entertained / fed elsewhere
As for the issue of the 14 year old… and whether you wish to include them / consider them an Adult. Well that is up to you.
As the Host of a social gathering, YOU get to decide what type of event it is you are having, and who the Guest List will and will not include… and where that cut-off point is… 14 and above, 16 and above, 18 and above, etc.
My main advice, tho is WHATEVER you choose to do in regards to this issue with kiddies, that you (a) are consistent, and (b) you get the word out to the key players (Both sets of Parents, and the Bridal Party)… so EVERYONE knows ahead of time and is on the same page…
Because, I can assure you that there will be Questions raised… WHY ? WHY NOT ? And HOW COME ????
And a word of warning… if you back-down for one, you can expect the whole thing to come tumbling down around you in utter chaos, as others try to wrangle their way as well.
Hope this helps,