The first one is 'Pink Diamonds' and the second may be 'Peace' (if it was large I would say 'Mikado'). To keep them you might try cutting a couple leaves off each one and propagating them to have some back up insurance (that is what I do now after losing bunches over the years).

The one I showed above was bought at Harmony Foliage with Mr. Begonia (last year's ABS president) telling me the names as I was labeling everything. What can I say? I just go with the flow... That seems to be the problem with names in begonias (and everything else) - someone drops a tag and sticks it in another pot or you can't read someone's writing or everything is labeled rex begonia. If only someone would come out with a definitive book on (fill in the blanks here) Check out the international database of begonias for Muddy Waters. Ignore the 3rd photo since it is the one I just posted above. Begonia IDB of Muddy Waters

latest report :). my all newly bought plants did very well for about a month and then i started tenting several in tall sleeves as i noticed that new growth was drying up at the edges a bit. then around jan 25 i noticed dreaded mildew spots on 'Pandora' and then after 2 weeks on 1 leaf of 'Red Raspberry beret'...i had no fungicide :(...so i removed offending leaves and sprayed with a little soap and peroxide in the solution. and then after about a month i sort of dunked them upside down into very light soap/peroxide solution for the transfer home... the trasfer to cold north was kinda brutal on the largest ones - leaves got crushed here n there and they were dark and moist for a couple of days...then i sprayed them with fungicide finally at home and tented all. so the long story short 'Pandora' lost most leaves and is barely alive, but has 2 new leaves coming clean :). it was a good mid-sized plant, strong too. so should pull thru. 'Fire breathing dragon' was my fave, but it was the smallest plant and only 1 rhizome in a pot, it stopped producing those lovely curling leaves even in FL, all new resemble 'Ring of fire'. It lives, but with only 4 small leaves (lost 4) and new growth tip dried up. so waiting now for a new bud to show up, hopefully. "red raspberry beret' was a Big plant, but sev leaves got ripped up and bruised(as i expected!) and i removed them...but it has sev growing tips and is ok. 'Xmas eve' did the best - it was tented from the start - so no mildew! and no damage of any kind. But it had smallish dense leaves and produced a good amount of new growth. it's nice and bushy even now...it's not that impressive, not much pink in the leaves yet, it needs to mature some more for that i guess. so may be i'll manage to click a few pics soon. then and now for comparison.

Thanks Laurie. I have had her positioned in a better lighted position since this picture was taken, but no signs of blooming yet. I noted you said you thought yours wasn’t blooming because of too much light? Is that typical? Am I correct to understand that they bloom quite easily and generally late summer or fall, or can they bloom at any time? I wonder because I have a beautiful Gryphon Begonia which I understand is related to the cane types, and it blooms when the daylength is quite short. Mine bloomed for me in February of this year ( indoors of course) after being outdoors until close to my first frost( late September here).1st pic blooming in February. 2nd pic as it appears outside recently. Just thought I’d share that, hope you don’t mind.