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In English it's simple "let's meet" or "let's meet up".

In Spanish, even after having done a research, I can't figure out which one is correct and appropriate for an invitation to a date.

1. nos vemos




2. vamos a encontrarnos




3. vamos a quedarnos




4. vamos a reunirnos




5. ???

Which? When is used each one?

Note, my question is not how to say "let's have tea together" and not "let's go out".

For me, it's clear that the verb is Quedar, and the usual sentence is

¿Quieres quedar para ... ?

Liek for example, Do you want to have a tea together? You'd say

¿Quieres quedar a tomar un té?

or

¿Quieres que quedemos para tomar un té?

Or just with quedar:

¿Quedamos para tomar un té?

The imperative "quedemos" sounds kind of "too formal", it's better to say it as above, as as suggestion.

This one works out with friends and with relationships, so it's very widely used, at least in Spain, where I live. Any friend must frequently ask you one of those sentences.

It is also what you say after being introduced to a girl (or boy) you love. When you've already talked for a while. You'd use those sentences to invite her (or him) to have some drinks, or to go somewhere, or whatever.

In sum, it's definitely the one you're looking for.

I think that the translation will depend very heavily on which version of Spanish do you speak. For example, I am Cuban, and there it is not very common to say "Vamos a quedarnos" or "Vamos a quedar". In the island is more common to say "Vamos a vernos". But aside from which translation is more used by region, any of the following is perfectly correct and very neutral (in the sense that you will be understood in virtually any Spanish-speaking country):

1) Reunámonos / Vamos a reunirnos.

2) Juntémonos.

3) Vamos a vernos / Veámonos.

4) Vamos a encontrarnos / Encontrémonos.

I wouldn't use "Nos vemos" because that's like a farewell in many countries, and you don't want to say Goodbye, you just want to arrange a date of some sort.

En Colombia es común escuhar la expresión "hagamos una reunión", como también "vamos a reunirnos".

La reunión puede ser presencial o virtual (por ejemplo usando herramientas comos Skype) o mixta.

Ejemplo:

hagamos una reunión para fijar las fechas de entrega

vamos a reunirnos para discutir el alcance

I can't figure out which one is correct and appropriate for an invitation to a date

Maybe it depend on how you are meeting the other parties. I would rather use"reunámonos" than "vamos a reunirnos"

Reunámonos el viernes para tratar este tema.

I think this one doesn't imply how you are meeting (like, in person or using Skype, FaceTime or the like).

"Veámonos" could imply to meet physically, but not necessarily. The context could imply you meet online. I would not use "Encontrémonos/encontrarse" because this one could imply "to meet in person".

Encontrar

5. prnl. Dicho de dos o más personas o cosas: Hallarse y concurrir juntas a un mismo lugar.

You could understand that this could be a virtual space, but the first meaning of encontrar (to find), still carries strong connotations with this word. After all, you don't say "let's find each other somewhere" as you say "let's meet each other somewhere".

The option "quedemos/quedarse" could be another good option to imply "to meet somehow".

I don't know about other places, and I'm also not certain this is still up to date, but some years ago there was a standard code phrase for this:

¿Quieres tomar un café?

or

Vamos a tomar un café, ¿sí?

If it's vaguely in the future:

Vamos a ponernos de acuerdo para tomar un café un día de estos.

It was analogous to the opening move in a chess game. The invitation for coffee was the opening gambit.