I think it's a mistake to try to find the cause, and to try to dismiss her worries and stresses by saying there's no reason for this. You kind of argue both sides here: you say she has anxiety and worries about school/friends/boys. But later on you say you have a wonderful family and there are no pressures about anything.
My husband's ex kept writing off both kids' problems by saying she was perfect and the only problem was my husband. So she didn't get them treatment. They are both in abusive relationships and taking anti-depressants and unable to deal with pressures without running away. There are many causes of anxiety and depression - many of them are chemical and not based on external forces like peer pressures.There are also fluctuating conditions that manifest as mood swings - so an outgoing and joking person can be quite desperate at other times.
I'd talk to the pediatrician immediately and find a counselor. When your daughter is at school, make a copy of the relevant diary pages and show them to the counselor. Get advice on how to approach your daughter if you can't figure out a way to get her to counseling otherwise. I do think it's okay to say you had a nagging feeling and you want to help yourselves get through high school with better communication (not all the details, of course) and to work out strategies for handling the stresses and difficulties that may come along in high school. But you can expect pushback if she thinks the only one who needs help is her.
I think teens/kids have a reasonable expectation of privacy in general, but I also think parents have a right, and an obligation, to check their phones, browser history, dresser drawers (for drugs, weapons, evidence of sexual activity) and even a diary. You didn't read it just to be nosey - your intuition told you something was off. I wouldn't tell your daughter right now, and I'm not sure how much I'd involve the school counselor (depends on how much attention you feel they can give). It may be harder for your daughter to think "the whole school" knows about it.
Can you say why you think you wouldn't tell her dad about this?