Advice on cold notes splits into "be genuine" and "paste this script." Neither helps you write the next one. Here is what each line is for, so you can change the words without losing the point.
Line 1: who you are, and why this note exists
Identify yourself and the role in the same breath.
"Hi Priya — I'm interested in the Senior PM role on your growth team."
Not "Hi, hope you're doing well." The first line is the one line that gets read. Do not spend it on weather.
Line 2: a reason that is actually specific
Give a real link between your work and this role, or this person. Not a claim that would fit any job.
"I've spent four years on onboarding, and the part of the role about activation drop-off is close to what I've been doing."
Not "I think I'd be a great fit." An assertion with no detail does not change anyone's mind.
Line 3: a small ask, with a second door
"Would you be open to a quick chat, or could you point me to the right person?"
Two acceptable outcomes. A note whose only useful answer is "please refer me" gets ignored more often than one that lets them help halfway.
Line 4: take the pressure off
"No worries either way — appreciate your time."
Not "Looking forward to hearing back soon." That raises the cost of silence, which makes silence more likely.
A short note that is specific beats a long one that is trying to be impressive.
Peer versus recruiter
The shape stays. The emphasis moves. With a recruiter, line 2 should meet what the role needs. With a peer, line 2 should meet their actual work — "what it's like" is easier to answer than "please evaluate me."
You are asking a stranger for a small favor. The message should be the size of the favor.