in our case, DD8 had a test when she was 2.5 that revealed her to be "at least" 2 deviations above normal. Last years District administered tests confirm this but we are also going for private testing because the score was so high. We have had a terrible time with school and are leaving our current school...I would not have put DD in kinder at 4 (she turned 5 the second day), but wish we could have skipped first grade, which I now believe really damaged her. I believe she is at least a full grade level above her peers, especially if you don't have dedicated GATE classes and that if she had the same second grade teacher a year earlier, it would have made a WORLD of difference...
DD4.5 is similar to what you are describing with your second, social, outgoing, excitable, jubilant and full of energy, where DD#1 is more like "emo" (yes, it started at 2, btw). DD4.5 has not demonstrated the desire to read on her own, but we just placed her in an academic based preK/K (it will be our choice to place her in first or kinder next year, she will be 5.5 on the first day of school) and the teacher says she knew all the words they wrote on the board and that they counted together to 100, DD was able to give me a full account of her day and had a BLAST doing the homework, which was very similar to what DD#1 did in public kinder.
If I were you, if you have the resources, test privately and start looking at enrichments. I don't know how your school/District is, but our school is driving me up the wall, not recognizing that gifted is gifted, it doesn't start at 3rd grade and I think my DD#1 picked up many bad habits and inappropriate coping mechanisms, especially in first grade, when what she probably really needs is a more stimulating environment. I think the sooner you find out what she needs, the sooner you will be able to give it to her and that will make life better for everyone
