嬉しくて泣きそうになったって英語でなんて言うの？

I cried happy tears!- Happy tears means to cry with joy!

After speaking to my friend overseas I cried tears of happiness- Tears of happiness is similar
to tears of joy- a variation.

Usually, when you cry because you are happy, you state that you are crying tears of joy. Instead of using the word "cry", you could also use "weep". For example, "I was weeping with joy!"

A message received from an absent friend may be very moving - and even cause an effect which we may not have anticipated. it is always pleasant to receive a message from someone far away and to be comforted by the fact that they care about you and are thinking about you.

In British English, we use the phrase 'brought to tears' to describe a situation where you felt happy or sad and almost cried, but did not actually cry.

When I saw my friend for the first time in a long time, I cried tears of joy. She asked me why I was crying, was I sad because I missed her? I told her no, these are happy tears. I was so thrilled to have her back that I couldn't help but to sob.

examples
" it was so great to speak to you, it made me really emotional".
or
"i cried tears of joy when i saw her".
or
"i nearly cried i was so happy".

When you want to express how happy it made you to hear from someone via text, then you can say:
-I was so happy to hear from you, I wanted to cry.
-I was so glad to see a text from you I almost cried.