To wonder if these comments were rude

[Eggandcheesemuffin]

It's in a nursery setting, I do some interim work there. I am fairly quiet and can be shy, but I still do make some conversation with the staff and believe I can build good relationships with the children.
Anyway many of the staff there have very extroverted and outgoing personalities, though a couple are quieter.
I was in a toddler room I've worked in several times before, and the room leader, who is very extroverted, was telling a little girl off for climbing.
She then said, "Be careful, as egg and cheese will shout at you, and she's got a very loud voice I've heard it before."
I've never shouted at a child or used a very loud voice. Anyway I didn't really know how to react or why she said that.

On a separate occasion they randomly swapped me and another agency worker between rooms (they like to do that if it suits them).
The room I was moved from was very boisterous, and I was put into the calmer room which was fair enough. The said room leader said in front of everyone, "Sorry eggandcheese, you're good, but Katie (who they swapped me with) is just a bit stronger."

Ok, but why say that in front of the whole room? How am I meant to react?

The first sounds like a joke rather than being rude.
The second is a bit rude, but many adults don’t count children as people when saying things and so will act like they’re talking in private when talking to another adult. I don’t agree with that, children are little sponges and they certainly understand and remember what is said around them.

The first incident is poor behaviour management. You don't threaten a child with discipline from another person, or with shouting. She doesn't climb because she could fall and hurt herself. You should have took over and said that you wouldn't shout and explained why she should be careful, then addressed it with the other staff member later on. If behaviour management is lacking, take it to the manager.

The second comment wasn't ideal, but in some settings staff placing decisions are shared. If you feel the need to then bring it up one-to-one. If she means that you aren't assertive enough, then she should be giving you that feedback. My DD was a NN and one staff member's confidence was low because she was always put in the baby room. It was because she was so calm and soothing. Staff appraisals should be happening so you can address and work on any issues.

Try not to take things to heart op,it comes across that you are shy and sometimes some people will exert their personalities over yours.
Perhaps work on your self belief and confidence and remember nobody is better than you.🌈