I don't think it is Booster, at least not on a live chat program; like Kazuma and Sue were earlier. I think something more mysterious is happening here
Well first off, as pointed out; the cabin is in a ridiculous spot, no telepad; and the notebook covered with dust probably hasn't been touched in a long time...
Meanwhile Booster who fell nearly to his death in the labyrinth, probably has been busy himself, surviving; tuning his Booster v0.8 to 2.0; fixing the teleporter to escape from the labyrinth...
And the messages Quote recieves are very odd if we assume they were from Professor Booster...
"Why don't you take a rest?"
"You know, I often recorded what I learned in that notebook..."
"I pray for your victory."
Even if we assume Quote secretly types something to alert Booster to the situation, it would seem like Booster would give you more of a heads up on the situation with the flowers, with everyone being seperated and in trouble.
Keep in mind Cave Story has intelligent robots, so why couldn't the computer in the cabin be connected to some sort of intelligence?
Maybe it is connected to Booster in the past, or maybe something else is going on here...
"04-08-08 Wife bore child."
And as for the mysterious dates ..! I can't make heads or tails of those

If I take them literally in-game context.. hmmm well it's the only date we have to try to figure out the year and time that Cave Story is taking place... If it's an easter egg about Pixel well .. hmph! lol Guess it doesn't mean much ingame... Can't tell, ask pixel!
According to me, in Cloud Story, I'm possibly explaining the date through Dad aka, the mysteriously missing Sakamoto, never hinted about in Cave Story...
I mean why would all of the Sakamoto's go to the island? They all seemed to know quite a bit about the island... So did they have a history with the island? or were they just a few "lucky" crazy scientists who were allowed to go to the island? Don't forget 10 years ago the island caused the "War".
"Please heed my last request.
If you see Sue again,
I want you to take her and flee this island.
That girl is Sakamoto's daughter,
and she never wanted to come here in the first place.
But we couldn't leave the child all alone,
so she had no choice but to come along with us."
Also remember the Silver Locket?
[A silver locket in the shape of a fish. From the condition it is in, it looks like it's been treasured for years.]
"Oh!
That pendant!
You found it for me? Sue gave that to me!"
Also remember the words on the chest in the Mimiga Town?
"So long as you retain your spirit of exploration, surely you shall find your way out. This I believe."
Strange...
Hmm.. Also if you want to believe Quote lost his memories perhaps on the hard path he begins to recall them? You know the ominous text :
"Did you know that the witch-woman Jenka once had a brother?
His name was Ballos.
Like his sister, he wielded powers far beyond those of mortals..."
I mean otherwise we just have some crazy narrator who only voices the hard path stuff

I'd like to think there's a ton of mystery left in Cave Story ..!
I kinda got off topic but oh well!
Oh and finally some wierd things.. like Booster meeting Jenka... in the credits..?! At face value we can call that humourous lol
But perhaps there's a history and a reason...
Same with Booster's recognition of Quote "!!!"
I just have to say there must be more history we need to hear or tell through mods
*Cough* I have more but going back to the original question :
Is it Booster?
Maybe. But, I don't think it's him in the present at least...