While there may children who are 21 months old who can swim like fish and do the float underwater thing like the neat videos show... my son, currently 5-1/2, was NOT that child. And I fretted over it, because I had not had much experience in the water growing up and no lessons and thus am a weak swimmer myself, who hates to get water in my eyes or nose. I desperately do not want to pass that hesitance in the water to my children so I tried to start him young, too, with the mommy-and-me lessons... it didn't go very well at all! We had very similar difficulties (but involving ME, alas) and some resulting toddler-defiance on his part. He still liked the water for the next couple of summers but resisted going out with me or any further than the steps for quite a while, then ALWAYS had to have his swimmies on.
But, miracle of miracles, last summer he saw another little boy who was 3, a year younger than him, throwing himself into a hotel's hot tub with abandon and LOVING it. That marked the start of his "fish" days. Since the water was warm, we got rid of that obstacle ("I'm cold, mom!") and the example of a little boy who was unafraid and having fun encouraged him greatly (instead of mom just "pushing his boundaries"). Very quickly that week he progressed from swimmies and the shallow end to no swimmies, doggie paddling and jumping into the deep end! (You have to understand this was a HUGE big deal.)
Throughout this journey, I should mention, we provided plenty of toys and a variety of floaties throughout his various comfort levels, played games, offered incentives (this summer, he earned diving sticks for touching the bottom of the shallow end and now he's an avid "diver"), let him wear goggles (but also required that he not wear them all the time), etc., etc. I think the biggest thing for him was just that last summer and this summer, we have had ACCESS to a pool more than ever before. And the weather has been nice and the water warm where we've been (Florida and Houston, lots of hotels as we travel with my husband for business). Also, hot tubs. I limit the amount of time they can spend there, of course, but for my son when he was a little younger and now for my 2-1/2 year old daughter, the hot tub is nice to warm up and play in more comfortable water and depths (when adults aren't enjoying it and when it's not unbearably hot for the kids).
So, all this said, my son started to love the water at age 4, last summer. Then, much to my surprise given my adjusted expectations, my oldest daughter loves it this summer, at age 2. AND my youngest daughter loves it now at age 1 and just giggles or shakes water out of her face. We took her to the ocean several times this past month and she crawls straight out there, gets knocked over by waves, tumbles, gets water in her face and nose and just gets back up again and keeps going. (Her fearlessness can actually be a little worrying for me... I really have to keep a close eye on her! But I love it at the same time.) I think my girls have had an easier time with the water than my son for several reasons (maybe personality is a factor). First, they have my son to look up to and all they see on his face is pure enjoyment plus he is a very sweet encouraging child and always wanting them to play with him. Second, we've had access more often and to warmer pools/weather than when my son was younger (and we had to go to the YMCA or such). (Also, my youngest daughter got to be introduced initially to the water at two pools with a zero entry end... perfect!)
Third, I think I learned better not to "push." I had these expectations for my son and waffled between "he can do this!" and "be patient, he's not ready." I was a first time mom and didn't know WHAT to do and didn't want to let him down by underestimating him. But I think he came to dread pool times, because we'd clash, the water wasn't very comfortable temperature-wise and he didn't entirely trust that mama was going to honor his wishes. And while I was, of course, keeping him safe, I'm sure he sometimes didn't FEEL that way ... like I'm thinking your daughter must feel about that instructor, putting her underwater like that. I guess I thought that if I acted like that sort of thing was no big deal despite his tears, my son would think "oh, this must be okay..." but instead he just started to fight me more and not trust me when I'd say he'd be okay. He knew I was going to try to get him to do something outside his comfort zone, even if I didn't MAKE him, I was going to PUSH for it.
I finally have learned (and am learning) to be more patient with each of my individual children's learning styles, abilities and fears... there's a fine balance between honoring those and encouraging them to step beyond them. I just don't think it can be forced and I really think you have to protect their trust of you. You can explain and show and comfort that the water is okay and that she'll be okay... but I wouldn't let the instructor do something to her that she doesn't like. Honestly, from my personal experience so far, I'd say just get a pool membership and take her often to play, with you and other children, not have "instruction" until she's comfortable in the water. We're going to get my son lessons this summer, after we move, so he can start to learn different strokes. But he's finally completely comfortable in the water (spends more time under it than above it!) and his ability to understand and follow directions from an instructor is LOADS better than it's ever been before.
So, anyway, that's been my experience and that's my advice. :) Remember, you know your daughter better than the instructor and you care for her more than the instructor ever could. Trust you instincts (as I didn't) and protect your trust relationship with her first and foremost. :) And have fun together. If it's not fun, don't force it. If you can't get your money back, just play in the water together during the lessons doing what she's comfortable doing. Those mommy-and-me lessons we took when Hayden was 2-1/2 were some of the WORST experiences for me -- and I'm sure for him, too. Don't let that happen to you, too.
Blessings!
PS-- oops! Just saw your "what happened" note. Wonderful! I'm glad you got out of those lessons, into better ones, and I KNOW how blessed your heart was by your daughter leaping into your arms. Way to go, Mama! :)