Mom is 75 and daughter is 27. Married late or not (to second husband or not), that means she had your gf when she was 47-48 years old. Pretty unheard of even right now because of the risk to mother and child, but still biologically possible. Barely. I estimated your ages based on a much younger possible age for her parents on the basis of common sense. Women are afraid of having kids after 35 or so because of their biological clocks (causing birth defects, having menopause).
Screw trying to learn pat phrases. If Dad likes you as much as you say, you're halfway home. If Mom is more hesitant than adamant (what you wrote initially), then that's a softer target than you first told us. Help keep us clear what is happening here, or you will get different advice.
Just be your admirable, amiable, polite Ohio self. I don't know how you communicate with Mom and Dad on other matters, but treat this with a bit more caution and respect, but don't try to go into keigo and traditional poses. That is out of character. Let your gf do any explaining necessary if you flub it. You intentions can be read in your face and eyes and in the tone of your voice.
As for "lying" about where you will eventually live, don't. It's an honest answer to say you don't know just now. It sounds as if you truly want to stay in Japan for a while, so focus on that and whatever job situations you both have. Everything else is "We don't know". In my own case, my wife is the oldest child and the only daughter. She looks after her parents more than her brother, even though they live by themselves and are self-sufficient. When one of them dies, I'm sure we will somehow change our living situation to be in the same house with the surviving parent. This has been discussed between her and me a little, even though her parents are younger than yours. I don't know you gf's feelings on this, but it might be a factor in Mom wanting you to stay. I suspect not, and that the main reason is that she is the baby child. Keep all that in mind. Best of luck.