H.W.
Three IS often harder than two. Your daughter is a literal child-- most are. You said "I can't carry you because I've got these other things" so she modified her expectation and waited for you. She did try, but it wasn't the result you wanted. Be more clear next time in saying "I want you to come downstairs with me now". No 'reasons' or 'explanations' because they can often clutter the conversation. In this case, your daughter was focused on accommodating your need to carry things and so she expected to be carried afterward.
I could write a book on not letting your child's tantrums control your actions or how you parent. I have seen one family in particular whose child runs the show from wherever the child sits, screaming. The child has the parents to the point of nearly-infantalizing them because if they don't like what's going on, all the kid has to do is scream and cry.
Interestingly enough, this child did not do that with me because I would put the child in a quiet place and tell them firmly "you must sit here until you are finished screaming. When you are ready to do X, come and find me." Then, I walk away. No negotiating, no trying to talk the child into doing what I ask, nothing. This kid was a screamer with me at first-- would just sit on the floor and scream for me from across the room-- because the child had everyone at home trained. By our second day, that had stopped.
Kids who tantrum for attention or to control a situation need us to do what seems impossible: to turn a blind eye and LET them rage in a safe place. To be consistent ("You need to walk down the stairs now. I'll wait for you down here." and then sit at the bottom stair and not look up until she's come down. Or take her back inside and put her in her room. "Come out when you are done crying and ready to go down." ) Staying emotionally neutral--at one of the most difficult times to do so-- is going to be your best tool. If kids can see us getting upset and hooked in, they will draw the drama out longer.
Here are some things I consider before getting to a tantrum point:
1. Don't give a direction you can't enforce. For example, if you are wanting to avoid a tantrum in public, then that might be the time not to dig your heels in, but to instead anticipate how things *might* go. Then, this might look like:"Do you want to walk down with me, or would you like to wait so I can carry you?" If the stairs were, say, at the zoo, this would be more helpful than giving a direction she typically declines. There's nothing wrong with picking your times to put your foot down and deciding that this time isn't one of them.
2. When you do decide you want her to follow through with a request like coming down the stairs, be prepared that she'll protest. Take a magazine or something with you for YOU to do at the bottom of the stairs. If mommy's not watching her or getting upset, suddenly the fact that she's at the top of the stairs and you are not is HER problem, not yours.
3.Thoroughly consider her request before you decide yes or no, and then follow through. This will save you a lot of trouble, just a minute or two to think about it. Once you have said "no" to whatever she's asked, then that's your decision and you stick with it. It's one thing to change your mind if she's being agreeable and you realize "you know, I do think it's okay", but if she's tantruming, screaming or crying to get what she wants, this is not the time to say yes to make her stop. This is how parents 'train' their kids to manipulate them-- they let the kid think it works to cry and scream. It shouldn't work.
4. Remember, some of the things that come up are problematic with how WE parent, and some problems are the child's. If a child does not want to be safe, follow family rules/social rules (safety, taking turns), or otherwise comply with what's the 'norm', then that is *their* challenge to master their emotions and anger. We do better by not butting in and trying to make it better for them. If we get ourselves into a spot, we have to recognize it (so as not to make the same mistakes-- just like you did by posting this) and then figure out how to move through the moment. But it's good to know what's contributing to the situation.
Good that you are asking this question, sorry it's so long, but tantrums are rarely a "just do this" sort of thing. Little ones will tantrum for two reasons, primarily: either due to frustration ("my blocks keep falling over") or to influence an outcome ("I want you to pick me up"/"I want some ice cream", etc.) Know the difference and treat them appropriately. With consistency, it does get better. It takes time, but it really will. :)