Lose heart

I have never heard of the second definition. As a native American speaker I would never use the phrase "to lose heart" to mean "to fall in love". In fact, if someone used this phrase that way, it would confuse me.

It's "to lose one heart's to"

I lose my heart to her..

Hmmm.... I suppose. I would never use that phrase, but you are right. I would understand it.

but ...i have not gotten my answer yet..
i lose heart to her .
i lose my heart to her ...
they have 2 different meanings ?

Neither of those phrases are things that I as an American speaker would ever say. They sound unnatural. The first one doesn't any make sense to me (and has no meaning).

The second one might be used poetically (meaning to fall in love). It doesn't sound normal.

Things I might say..

-- I wanted things to work, but in the end I lost heart in my marriage. (This means gave up).

-- After she cooked me arepas, I gave my heart to her.