Kids will complain about you no matter what.

When my kid was that age she complained that I didn't have a job like the other moms, and we never had any money to afford things like a cell phone and nice vacations like the other parents had.

You are letting this comment get to you because it made you feel guilty or something. Don't let it! You felt it was fine to check a work email while you all were watching a movie and that's the way it is.

If you start trying to alter yourself to please your 8 year old daughter, there will be no end to the amount of altering you will need to do.

Just tell her, "That's part of my job. That's the way it is." If she doesn't feel you are paying her enough attention otherwise, sure that's something you can look into. But don't let her dictate when you do and do not look at your messages.

Most things can wait. Can you put an out of office reply stating that all emails recieved after 6pm will be answered between 10pm and 8am the next day?
Then wait until she goes to bed.

Or, you can have a set 'work time' from say 800-830 each night where you leave the room and answer emails. All else waits

We don't get to choose what our children remember unfortunatly. She won't remember a fun movie with mom, she will remember 'mom on her phone". You need to figure out if you are ok with that or not and what you are modeling for her as far as work/family balance is concerned.

I would take a hard look at when you actually need to look and respond. I know some work culture can dictate very immediate responses, but this often just fuels anxiety and can cause more work in the long-term when everyone is responding so impulsively and not taking a pause to make sure they are responding in a matter that solves ongoing issues. Some of us need to stand up for work-family balance! I get email notifications on my phone with it on silent, but can look briefly to see it is something I need to respond promptly too. If so, you should be able to watch a movie with your daughter and eat dinner as a family, except for the case of real emergencies, which (without knowing your profession) should be infrequent.

I find it hard to believe that you truly can't disconnect from looking at your screen. If it's M-F during working hours, sure, explain to her that these are your responsibilities and that they help provide for the family. But evenings and weekend you need to figure out a better system. There should be some time during the week that you don't need to check your emails. Figure that out, and then stick to it. You need some boundaries, and good for your daughter for telling you that.

OP, there are good suggestions here. One I didn't see is for the times when you are off (I'm assuming/hoping you don't need to be tied in literally 24/7), can you plan something for just the two of you? An ice cream date out in a restaurant where you do not look at your phone once the entire time including there and back.

Yes she needs to know that you have work to do and it's not always conveniently timed, but this sounds like she plain old wants some undivided attention from mom.