It would be a good idea to get a dog crate, if only to use for a short while as you are trying to teach her. It may work to use the family room, but most dogs need a more restricted space to really reinforce the idea.

Most dogs that are potty trained will not soil in their "living area". When they are young and new to the household, their boundaries of living area are not defined to them yet. As long as she can get away from it when she is done, she doesn't see it as a problem wherever she goes and is no different than going outside. This is where the crate helps. What the crate does is to restrict her area so that she has the choice to hold it or to wait until someone lets her out. Most dogs will not want to soil in their crate and if she is respecting her crate and keeping it clean and dry, that is step one. It sounds like she already has that idea if she doesn't like when you place her droppings in the fenced in area.

Next, you have to work on the positive reinforcement. You will have to work with her and take her out to the fenced area on a lead. Praise her for going there if she does potty. When you take her out, give her a few minutes to go. If she is preoccupied or won't go, bring her in and put her in her crate. Repeat the process in a few minutes and praise her like crazy if she potties when you go out. Make it very routine so she gets the hang of it (same door, same spot in the yard). You need to physically be there when she is standing up from a squat so you can praise her immediately (not when she come back to the door). Do the same praise method for urine or stool so she gets the idea that you are happy when she does this outside.

If you want to teach her to learn to "hold it" until she is taken outside, I would not offer papers or pads as an option in the house. When she feels the need to eliminate, her choices are to hold it when she is in her crate or she should be taken outside in the fenced in area. Obviously you want to take her out frequently during the daytime hours (every few hours and after eating). Take her outside before bed and first thing in the morning and don't ask her to hold it for too long at first (6 hours max). If she soils her crate, which most dog don't want to do, don't scold her and maybe take her out a little sooner until she builds up to a longer time period. Until she learns learns what the pen is for, I would make every visit outside start with a visit to the pen with you and her on a lead. Once she has eliminated, then she is free to go on her walk, play, etc. If she doesn't, take her in, use the crate (so she doesn't soil in the house) and try again in a short while. She will soon learn that she is praised for the behavior of eliminating in the pen.

Make sure the size of the crate is right. You want to give her enough room to lay down and stretch out comfortably, but not too much more so as to be able to soil on one end and get away from it.

One other factor with the bowel movements is to make sure she is not allowed to "free feed" or eat whenever she wants. It will be much harder for you to control BM's if so. At her age, twice a day is fine.

This is a difficult subject to cover in a few paragraphs, Here is a few link with more information:

housetraining

I hope this helps and please let me know if you have any further questions.

Her bladder control is likely good since she is a young dog.

As far as letting her go in the big yard or just the fenced area, it depends on what you want. If you only want her eliminating in the fenced area, when you take her out to it, give her a few minutes only. Don't let her putt around. If she doesn't go, take her back in. Don't allow her to run around or play in the big yard until she completes her business in the fenced area. When you take her in, watch her, because she probably has to go (that is where the crate helps so as not to allow an accident). In her case, if she is used to holding it, it may take awhile for her to learn this, but eventually, she will have to go. If you keep taking her out to the fenced area, and not allowing her to eliminate in other places, she will go at some point. Maltese dogs can be stubborn.

Ok, don't force her. If she goes out and lays down, let her lay (on lead) and if not urine or stool in a few minutes, she obviously doesn't have to go now. Take her in.