welcome to Mamapedia.
I too used to struggle with depression, so I understand how this disease can appear to others to be a "lack of motivation" and how it can undermine your feelings of competence.
I think you should expand your definition of "motherly" - it doesn't just mean a snuggly person. There are lots of great mothers with difference personalities, different degrees of comfort with snuggling, and so on.
Like the others, I am concerned that your 2 year old is in an academic setting with homework! That makes no sense. She's not paying attention to you because she is only 2 and she is not ready for "homework". Most 5 year olds don't get homework either. It's a developmental stage, not a personality quirk.
The vast majority of early childhood experts recommend a play-based preschool experience, usually starting around age 3. The purpose of preschool is to give children a chance to interact with other children, to begin to understand the basics of separating from Mom and Dad to spend time with a teacher or caregiver, to begin to participate in a group setting (circle time, story time, snack time, craft time, outdoor playtime, etc.) and to transition from one activity to the next along with the class. They learn (gradually) to negotiate things with speech rather than with physicality ("May I have a turn with that toy?" rather than grabbing it). They learn to hold a crayon and do some painting (fine motor skills) and they play outside on play structures or do other things for gross motor skills.
These are very basic skills and the only purpose of preschool. If a child learns to write her name, that's a bonus, but it's not necessary. I cannot imagine what your daughter is trying to absorb through "homework" and why it's important. That's a sure way to get her to hate learning. She doesn't need to learn any quicker - she needs to be allowed to develop in her own way. She may not have a lot of words but you cannot force her. Meantime, she's probably doing other things earlier than the other kids, so look at her for her uniqueness and marvel at what she does do, rather than measure her against some artificial yardstick. Kids learn to talk, walk, feed themselves, read, potty train and sleep through the night all in a different order. There's no "right" way.
I think you need way more support than you are getting. I think you should speak to your doctor about your depression, find a counselor who can help you with your feelings of inadequacy (your doctor can refer you), and see if your town offers some other forms of enrichment for your daughter and you. For example, see if your town's department of families & children has some parenting groups (classes or just play groups & support groups), see if the Newcomers' Club or the Y have play groups, check with the public library on story hours for young children, see if local churches have simple play time and mothers' networking groups that have nothing to do with teaching kids anything about religion or Bible stories but just community and togetherness and fun. A lot of play groups spawn friendships for moms and even some chances to trade off babysitting so you get some time for yourself while your child plays with a new friend. For me, sitting in those groups while the kids played and the moms chatted was a live saver - we shared tips, frustrations, fears, laughs, and more.
I think your fiancé could use some sessions with you in both counseling and a parenting group. You are very dependent on him and you have no other family around, and I think he could be more emotionally supportive of you while you get used to parenthood and have some time to work through this depression without him making you feel worse.
Yes, you need support - but you are not alone. We all do. We've all been where you are. We've even on Mamapedia for support too, but it doesn't replace the face-to-face you can get in your own community.