We don't know in great detail. Jesus' body, as it was ressurected, appears to have had some "interesting tricks" - like appearing in a locked room without using the door, materializing to whatever degree necessary, walking in the company of people that knew him, without being recognized, Travelling at will between Heaven and Earth, etc. As a matter of fact almost NOTHING about the "Life on the other side" is known. The "Mansions in Glory" imagery makes NO SENSE in a place where protection from weather isn't necessary, and there's eternal day. God simply states that "We can't imagine". So I don't waste my time trying.
Frankly, The concept of "Eternity" when seen from the strictly human limited viewpoint, isn't all that appealing.
I had the same thing come to mind, have always taken "spiritual body" to mean there is a glorified body that is not like the horribly flawed natural body that we can look forward to. As others mentioned, this involves not being any longer mortal or of flesh corruption. But, what's more, we can see in the resurrection body of the Lord Jesus that His body was not just a natural body. As you well mention, a couple facts speak volumes, that the resurrected Lord Jesus appeared in a room where the door and windows were shut: the Lord appeared, as if through walls, yet of a sort of spiritual to matter transformation, such that He could be touched and felt. Still, no flesh body goes through walls: there is obviously the spiritual body, that can also be controlled to produce a physical manifestation. I'd sure call this a spiritual body, anyway.
John 20:19 Then the same day at evening, being the first day of the week, when the doors were shut where the disciples were assembled for fear of the Jews, came Jesus and stood in the midst, and saith unto them, Peace be unto you.
John 20:26-28 And after eight days again his disciples were within, and Thomas with them: then came Jesus, the doors being shut, and stood in the midst, and said, Peace be unto you. Then saith he to Thomas, reach hither thy finger, and behold my hands; and reach hither thy hand, and thrust it into my side: and be not faithless, but believing. And Thomas answered and said unto him, My Lord and my God.
I'm not sure how many either realize that, when the angel rolled the stone away from the Lord's tomb, he said the Lord is not there in the tomb: this would indicate the Lord left the tomb, before the stone was even rolled away, to me meaning the stone was rolled away as further supernatural evidence, not, at all, to free the Lord from the tomb. So, that's more testimony that neither describes your Daddy's body!
That's what a spiritual body is to me, for what it's worth. Also, I'd add that there is scripture of spirits of the blessed in heaven, scripture the spirit returns to the Lord and such, indicating we don't really die, even before the first resurrection to a body that is like Christ's. Scripture states we have a purely spirit self that goes to be with the Lord, before the first resurrection,
2 Corinthians 5:8 We are confident, I say, and willing rather to be absent from the body, and to be present with the Lord. I'm not certain, at the scripture dissection level, that is having a body, but I wouldn't object to the spirit that goes to be in the presence of the Lord being called a spiritual body, as it's a spiritual something, with image of God attributes, clearly. Also, I think the point of the verse is that there is the temporary and corrupt, then there's something much greater, holy and eternal, to attain to, that is, whatever you want to call it. On the other hand, "spiritual body" seems to best describe the body of the resurrected Lord, that has the power to, in all regards, be a body that can be touched, that is, both spirit and bodily attributes in its nature. But, again, whatever you call it, I'm with you, not to waste time seeking to quantify how many angels can dance on the head of a pin, or care for these running debates over what is is. The older I get, the more and more I realize Christianity is a state of being, not being stuffed with doctrine, engaging in senseless debates over what should be Sunday school, settled doctrines, or matters there are only plausible, speculative explanations of, at all, with many claiming they have the answer, and this all the time on the likes of message boards, answers to matters that have never been resolved, with certainty, in all theological history. Scripture is very clear on this, with regard to spiritual maturity, but let's not get into that, not here, heaven forbid! We'll have no talk of spiritual maturity, moving on to perfection, shoot, any talk of literal scripture truths a child should understand or talk of good behavior, for that matter! LOL!