Enlarge Steel Defender while riding

I am a DM that has a player using a Gnome Artificer Battle Smith riding his Steel Defender companion and he wants to cast Enlarge on the Steel Defender.  He believes it would Enlarge him as well because the Steel Defender is carrying him.  I know in the end this is going to be my decision, but I'd like to hear what the community thinks and why?

https://www.dndbeyond.com/spells/2619170-enlarge-reduce

These are the 2 lines that have me guessing "For the duration, the spell enlarges or reduces a creature" and "Everything that a targeted creature is wearing and carrying changes size with it."

I say based on the line - spell only enlarges a creature.  A creature to me means only 1.

Secondly, to me carrying doesn't really mean the rider but only the items and objects the Steel Defender is carrying.  

At the time of the request, I did not allow them to both be enlarged but I'd like to hear what you all think?

Also, what are your thoughts on the artificer still being able to ride the Steel Defender if its enlarge and the Artificer is not?

I'm divided about whether it works on both(I would also go with a no), but I would put forth this: Even IF the rider of an enlarged mount would get the size up, to - they wouldn't get the benefits on the weapon attacks or strength stuff.

Thanks for the input.  He is excited about being able to enlarge both, so, I think I will allow the character to do it if he up casts it using lvl 3 spell slot.  I guess he will have to wait until he is level 9 before he can do it though.