Great question! By including te, you stress that you are interested about the addressee's girlfriend. It's like putting extra stress on the possessive pronoun in English: Is YOUR girlfriend pretty, too? This has the effect of contrasting the addressee's girlfriend with someone else's girlfriend (the speaker's, or some other person's girlfriend).
If you omit te, as you suggested, you contrast barátnő ‘girlfriend’ with some other person who is pretty, too, like the addressee's siblings, mother, etc.
To show you how this works, imagine Éva and Péter talking to each other.
Éva says to Péter: Nagyon csinos a barátnőm. A te barátnőd is csinos? ‘My girlfriend is very pretty. Is YOUR girlfriend pretty too?’
It would be odd for Éva not to use te here, since barátnő has already been mentioned in the context and she wants to contrast her own with Péter's.
Now if Péter says to Éva: Nagyon csinos a nővéred. A barátnőd is csinos? ‘Your (older) sister is very pretty. Is your girlfriend also pretty?’
In this case, it would be odd for Péter to use a te barátnőd because the contrast is between nővér ‘sister’ and barátnő ‘girlfriend’, and both are associated with Éva, not with two different people.
Try stressing YOUR in the different English examples above and you'll see that stressed YOUR is like adding te and unstressed your is like omitting it.
What you wrote is perfect. But since we are all different, someone else might say it differently. I might have said it like this:
Ez érdekes. Én lány vagyok, így tudom ezt a szót ebben az értelemben (is) használni.
See, "tudom" (I can) is emphasized, I moved it forward.
Seriously, that's all.