First of all, I have definately been there! You started a bad tradition because it wasn't realistic to keep it going so you will have a tougher time breaking the cycle now. You can do it but you will need to state your intention (don't ask), create and display a united front, AND do not waiver....not for one second.
You stated your mom is your only relative, what about your husband's? Does that come into play at all?
For your son's and daughter's birthday...simply pass out invitations with the pertinent information (date, time, PLACE, etc.). If you don't do invitations, be sure to state this information clearly (do not leave anything to interpretation!).
For the holidays, simply tell you mom (and anyone else that it applies to) that this year you are starting your family's new tradition. We will celebrate the holidays at home. Could you go to you mom's before or after you have the celebration at your home or on Christmas Day? I understand that you open presents on Christmas Eve but what about Christmas Day? It is hard to give you a suggestion without knowing that so I will tell you what we do and you can apply it however you may need to. It is a bit long because I want to give you an idea of what we dealt with and how it has changed.
We open our presents on Christmas Day but growing up, we always opened presents on Christmas Eve (I lived with my aunt so this applied there and at my dad's. On Christam Day, I went to my mom's and opened presents there then. My husband's family always opened presents on Christmas Eve too but only those from extended family as they were all at the grandparents home. On Christmas morning, his family opened presents at home from Santa (grandparents from both sides came to watch) and then they went back to grandparents home for most of the day. The grandparents (the one's whose home they went to) were deceased when I ment him so the get together had moved to his mom's home which later moved to a different day. So my Christmas Eve as an adult had consisted of at visiting at least three different homes and trying to coordinate with the rest of the family's schedule...resulting in a lot of rushing and not much fun. His mom decided she still wanted everyone over on Christmas Eve, back on Christmas morning! My FIL's birthday is 12/26 so we were back again the next day too! I drew the line (as an adult I always tried to stay home at least until my son left with his dad). The first year my husband told his mom we wouldn't be over on Christmas Day she didn't seem to believe it. I said it in front of his siblings, they must not have paid attention. That morning, they kept calling asking when we were coming because MIL would not let them open the presents until we arrived. So everyone thought we had not told MIL and were mad at us for being sneaky about it.
Now we still travel to those three homes (unless we can arrange for them to come here which works for most except for my MIL) and exchange presents. On Christmas Day, anyone who wants to see us knows where we live. Otherwise, we coordinate a visit before or after the holiday. It isn't idea but we deserve to celebrate and enjoy our holidays too. My kids should be able to be home for Christmas...and it is my job to make that happen.