Ok here is the situation. There is this girl I like in my school. We are now in final year and there is this big group project for graduation. We happento end up in same group(1 group consists of 13 people ) . First time I haven't developed feeling for her,but after 1 month I started to fell for her then I started mailing her. The reason I chose mail because in school she would always hang out with her all girl group , and since I also not too good talking with 2-3 girls at same time.
The mail progress was good and we also have a nice quality tak and jokes at school, we have been mailing for 6 months and she always reply my back. At the 3rd month in that 6 months I began to show my affection in my texts, like for example there was one time shetold me in school that today she had a shift for her part time job, then after school I would mail her 'do your best in your part time job today', and she replied me warmly with 'thank you , you took all the trouble mailing me, I will do my best'. At this point since she always replying to me I told myself that maybe it is a good signal and also good time to move to next level, so I began to be abit more bold. Like when we talked in school and she seems to enjoy it I would sometimes pat her gently on the head to show my affection, or pat her on shoulder when greeting her in the morning. I am abit inecure with my own action though, I am afraid that my action would bother her private space, making her being awkward or something, but since she still respond to my text warmly I took that as a good signal from her .
Realizing that, 3 days ago I invited her for lunch after school, well we'll be having about 1 week winter holiday by the end of December, during this period our graduation project is busy ( the deadline is at march and it is quite short for our project scale ), I know it was not a very good time to invite her but I thought it would be abit 'forgivable' if the lunch were to be at one time before the holiday.
So I texted her for lunch thing, and surprisingly the reply came 3 days after ( she never done this before ). In her reply she said that her cell had been broken she said her apology to me, as for the lunch she said that she was happy I invited her but she doesn't know whether at that time she will be available or not she can't promise that,Plus this December she said that she asked her manager in her part time job to give her holiday just this December, which get her a little bit low on money. So I said to her it was okay I understand, I am also sorry I invited you suddenly during this period I mean our project was getting busier and I invited you without thinking, my bad . Let's do our best for the project.
. For an addition she told me about her part time job situation during one season of free talk in school before my invitation came so I think her 'money reason' was indeed a fact.
It is just I was wondering, whether her reply mail to my invitation was a 'soft rejection' or was it her being honest. I feel bad having this doubt though, but it is not wrong to be cautious. Please guys I need your opinion. And I am sorry if there are wrong grammar or spelling , I am still learning English

It sounds like she was being honest. She has a lot happening. Why don't you tell her that the invitation is still good, and you can find a time that works for both of you. If she gets a lunch/meal break at work, you can even offer to bring food and spend her break together. That way, you still get your lunch date, and she doesn't have to worry about missing work or time on your project.

It sounds like she was being honest. She has a lot happening. Why don't you tell her that the invitation is still good, and you can find a time that works for both of you. If she gets a lunch/meal break at work, you can even offer to bring food and spend her break together. That way, you still get your lunch date, and she doesn't have to worry about missing work or time on your project.

First , thanks for the reply hheath. I will follow the advice for the lunch thing, I'll just have to find proper time and courage for that though haha, during lunch break at school it is hard to initiate the talk because she always lunching with her 2 girlfriends, For an update, just yesterday I gave her reply saying that it was okay, and I apologized to her for inviting during this busy period, it must tough I said. After the reply, when I see her at school, it might be just my feeling but I feel she ( or maybe me ) was abit tense, abit relieved when she said 'thank you for the good work today' when school ends.