Generator help and questions for a newb

I suppose I didn't know as much as I thought I did about what to ask for when I consulted with an electrician to put a transfer switch on my house.

Here is what I have: (1) Generac GP6500 portable generator with a four prong NEMA L14-30R female receptacle (in addition to two regular 120v receptacles) and a 30A breaker switch (2) on panel where power comes in to the house, a Generac gernerator interlock installed between the main circuit breaker to the house marked 200A and a dedicated 50A breaker marked "generator" - I think I'm using the correct terminology (3) Adjacent to the main power box, there is a box with a 125/250V 50A 3 prong male locking inlet - I'm trying to be as specific as I can because clearly I don't know what the hell I'm talking about.

Question is this - (1) I think this is compatible, correct? 30A breaker on the machine going into a 50A circuit to the house - again perhaps a stupid question. I read the manual and with regards to the cord, all it said was cord should be rated 250V and 30A or greater - I took that to mean 50A or 30A cord with appropriate adaptors would be ok (2) Assuming it is compatible, I am going to have to place an adaptor either with a female 3 prong locking end to go to the new inlet at the main box and a 4-prong male end to attach to the female end of a 4-prong 30A cord which then plugs into the machine with its 4-prong male end or a 4-prong male adaptor to plug into the machine with 3-prong end to plug into a 50A 4-prong extension cord. Which setup is best or does it matter? Or did I get the wrong thing installed on the house and have to call them back out to change it?

Thanks for any help. I know I can call and ask the electrician, but its easier for me to anonymously look stupid on the internet than profess my ignorance to people I know, lol.