Cold email templates - Please critique

I have two email templates for two scenarios that need critiquing. I'm still trying to get a banking related internship (have other opps as backups) for the summer and there are a few boutiques in my area that I feel I could reach out to.

Scenario 1:

Found this person on LinkedIn, immediately became intrigued by their work, felt the need to reach out. The main problem here is I have no real connection to this person, other than the fact that they went to the school that I grew up rooting for. Also, I'm not sure if I should leave my resume attached.

Dear Person,

My name is [my name], and I'm a rising sophomore at the school from [hometown]. I am pursuing a double major in economics and mathematics and hold an interest in private equity and the capital markets industry. I'm inspired by your professional journey and would love to learn more about the work you do at [pe firm person works at]. Do you have some time this week for a quick call? If you prefer, I'm here in [hometown] for the summer and can meet in person. I have attached my resume for context.

Let me know if this is a possibility at your earliest convenience!

Sincerely, [my name]

Scenario 2:

Talked to this guy before and we had a great conversation, where he gave me a lot of tips (related to both banking and life in general). I have no idea if this firm even has an internship program (its a very small boutique), but he was very responsive the first time I reached out so I thought what the hell.

Dear Person,

My name is [my name] and I'm a rising sophomore at the school from [hometown]. We first spoke back in January and since then I've only furthered my interest in investment banking by applying and joining the [finance club]. I'm also pursuing a double major in economics and mathematics.

That being said, I am on the hunt for a banking/capital markets internship for this summer and was hoping you'd be willing to speak about the opportunities that [boutique bank] may have. I know that you are on a busy schedule so any time that you spare to talk to me is greatly appreciated. My resume is attached for reference.

Sincerely, [my name]