There are two sentences below. Can anyone tell me which one is correct or are both wrong? If so, what is the correct way to say them?
Happiness is witnessing a rain in summer.
Happiness is being witness a rain in summer.
By the above sentences I wanted to express "I am happy to see a rainy day in the summer".
One more thing, can I use "a rain" to represent a short period of rain?
I'll start with your last question first, because we need to modify your original sentences so they make sense first.
One more thing, can I use "a rain" to represent a short period of rain?
No, you can not use the indefinite article to refer to rain, as rain as you have described it, is not a singular noun but is synonymous with rain drops, note plural, or rain fall which also does not take the indefinite article.
The singular noun rain has a different meaning:
Rain
1.2 (singular) A large or overwhelming quantity of things that fall or descend.
‘he fell under the rain of blows’ (OLD)
Your first sentence:
Happiness is witnessing a rain in summer
Rewritten to make sense:
Happiness is witnessing rain in summer.
Or
Happiness is witnessing rainfall in summer.
Your second sentence:
Happiness is being witness a rain in summer.
Rewritten to make sense:
Happiness is being witness to rain in summer.
Being witness and witnessing are synonymous so both the modified sentences 1 and 2 mean the same thing.
Both however do not specify the period of time over which you witnessed rainfall to make you happy. It could have been a 1 minute shower, days, or even weeks.
So in fact given that:-
By the above sentences [you] wanted to express "I am happy to see a rainy day in the summer".
You would be better saying exactly that, which makes sense unmodified:
"I am happy to see a rainy day in the summer".
Or more idiomatically:
It makes me happy to see a rainy day in summer.
Note that the indefinite article works fine here because it is referring to the singular day which is described as rainy and not the singular noun rain.
From what I understand, 1 should be correct, possibly with minor modifications. Here, "witnessing a rain in summer" is acting as a noun. The way it is written, it can be ambiguous, with "is witnessing" being a verb.
Depending on where you are using the sentence, maybe you could write "Happiness is, witnessing a rain in summer." But I'm not fully confident about that either because of the comma.
