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...There was a strobe with a diffuser on an umbrella to camera right, and a strobe with umbrella to camera left, which was my key light on 1/2 power.
For most portrait subjects, I find that having a large fill source close to the lens axis works best, with the placement of the key light and the type of modifiers on the key light, if any, used to change the look.
It's often said that the fill should be placed opposite the key, but I find that the use of a large, near-axis fill will prevent dark holes under the eye brows, in the corners of the mouth (including inside the mouth in smiling subjects), under the chin, etc. much better than an off-axis fill will. You can still get dramatic modeling by controlling the key placement and the key:fill ratio, but it won't be as harsh as with cross-lighting from split key and fill.
...On photo one I was hoping to get a Rembrandt effect, but I don't think it came out as strongly as I had hoped.
Looking at the lighting, especially the catchlights in her eyes, I don't see evidence of a Rembrandt key-light placement; it looks more like Paramount/Butterfly lighting. For Rembrandt lighting, the key light needs to be fairly far to one side, and should be a fairly hard (small) source.
This might help:
...Number two is with my key move closer to the middle.
Look at the shadow cast on her face by the dog's head. The key light is way too low for normal lighting.
...Number three is with a diffused fill light to camera right and a key with snoot directly in front of the subject.
What was your intent? In other words, why did you choose a snooted light as a key light, and why the low frontal position?
That kind of lighting isn't neccessarily wrong, but it's a pretty specialized look that should be limited to situations where you have a specific goal in mind.
I agree with the suggestion above about experimenting with a lamp to see how light position affects the "feel" of a portrait. Although they're too dim to be a good photo light, a reflector flood bulb from the hardware store in an inexpensive fitting on a light stand can be a great teaching tool. You can move the light around the subject, change the height and tilt, and so on, and then return to the "camera" position to study the effect. A patient human model is handy, but lacking that (or not wanting to over-burden her generosity) a large doll or mannequin, a soccer ball on a table, or something similar can be very handy.
Here's the kind of lamp fitting I'm talking about:
And here is the kind of bulb I'd use in it:
Lastly, although "just doing it" is a great way to learn lighting, studying the classic patterns can accelerate the learning process since one needn't reinvent the wheel, and learning the standard vocabulary can help when discussing the topic with others so that everyone is on the same page.
Here's what I consider to be one of the better reference sources on the Web: