Landlord's response to request for something in writing sound fishy?

Good to meet you on Wednesday
As far as the 'terms' in writing, This is the basic idea.
The rent is due on the first of the month, Security deposit is
one
months rent, 1/2 due the first month, 1/2 due the next. If move-in date
is
not on the first of the month then the rent is prorated

At receipt of deposit I generally give the new tenant keys to the
house.
also give the new tenant permission to move 'items' over to the rental
house
before move-in date. It is expected the current tenants be given notice
one
day in advance if you are moving 'items' over, The . I have nothing to due with the utilities As for a move
out
date, I would like one months notice. I also have a 4 page written
lease
legal document. I don't use this document, generally, for this rental
property. In the past I have used it at the request of a tenant, for
documentation for a load agreement for the tenant.
1/2 the deposit will 'reserve' the room hope this helps

Please cite your sources hoping to have some lawyers answer!
PS Why wouldn't a landlord requires a lease or at least a written rental agreement. He already mentioned he has a document but seems reluctant to give it to me.

Not really, sounds like she spelled all of her conditions out for you.

He is essentially offering you a month-to-month tenancy with very generous deposit/move-in terms.

I would never be so casual with a tenant. His wording is a bit vague, which may be what makes you question him.

"the basic idea"
"I generally give the new tenant keys " and
"for a move out date, I would like one months notice"

These things should be set in stone. The only thing I would do in your case is ask that you use the lease document (because it will contain protections for both parties) and substitute "m2m" for "annual" if you want.

If that response was in an email, then you have something in writing. Not exactly a lease, but some details. He said he does have a lease document if needed.
Is this a room rental? If so then you may have a more of a boarding house type arrangement.
A lease does give you certain protections, and oblige you to certain standards/rules.
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What sounds fishy?

As long as you are not paying by cash, & you are getting a receipt for your deposit i dont see anything even remotely fishy