No, @BrownParent is right - either that or your question is not worded in a way that accurately conveys what you want.
There’s no answer to the question “how, exactly, [do graduate programs] “choose” which applicants have “better” research experience.” That’s because graduate program admissions committees are made up of professors, each of which have different ideas about what’s “good.” In engineering it may be a single PI or two that picks you, so really, your admission is more dependent on how that PI judges you.
But BrownParent is right in that no one is going to care about your adviser’s writing ability, nor is it as simple as parsing the difference between “this kid is a genius” and “this kid is the best researcher in his class.” Both are glowing statements that are likely to go a long way towards convincing a committee. The name of the position also doesn’t matter - it’s almost always “research assistant” anyway.
What matters is
-What you do there. What are your tasks, and what do you know? Cleaning pipettes or rat cages is different from recruiting participants in studies or setting up experiments, and setting up experiments is different from writing literature reviews and presenting at conferences. The ideal set-up is a progressive journey of experience, where you start out doing more menial tasks and progress to doing more research-related tasks that will prepare you for graduate school. The way you showcase this is to discuss it in your personal statement and talk about it at interviews.
-Quantity/length does matter. Doing research for 2 years is better than 6 months, and doing research 15 hours a week is better than doing it 5 hours a week. Someone who volunteered 15 hours a week for 1 year still has more 50% more hours in the lab, and thus 50% more time to learn skills, than someone who volunteered 5 hours for 2 years. Five hours a week, to me, signals cage cleaning and literature searches and not the more substantial work. I think
-How your advisor talks about it in the letter. But the difference is not as trivial as the difference between “genius” and “best researcher.” It’s more like - what did YOU do in the lab that gives your adviser good things to say about you? Did you always show up on time and were you reliable? That’s basic. Did you complete all tasks as assigned and turn out good quality work? That’s good. Did you go over and above the call of duty, propose new ideas for students, present independent work at conferences (student, regional, or national), help co-author a paper, or something like that? That’s the kind of experience you WANT to have so that a PI can talk about it.
But you can’t worry about your PI’s ability to discuss it; the only thing you can do is do excellent work in the lab.
-Fit. The research that you do doesn’t have to be exactly related to what you want to do in graduate school, but your research interests should align with the department and you should have some sense about how the skills you learned in your undergrad lab will be transferable to the work you’re going to do in your graduate program.
The other thing is that depending upon the type of master’s program you want, professional experience (internships and part-time jobs) might be weighed as or more heavily than research.