To some extent this question is asking for an opinion. If someone said to me
2 "I am short of money"
I would think they have insufficient funds to meet their current financial commitments, rent, bills, food.
On the other hand if they said
1 & 3 "I am a little / somewhat short of money"
I would understand that they probably had just enough money to cover their current debits but nothing to spare for a new venture / expense.
1 "I have less money in my account"
needs qualifying. On its own it is ambiguous. It should be something similar to
I have less money in my account [than I thought / than I had last week / than
you / after paying all my bills].
2 "I have to make some expenses, and I exceeded my budget"
Is not quite right and again ambiguous. A BrE speaker would probably say
"I have to meet some expenses OR I have to make some payments and I have exceeded my budget."
They may have exceeded a project budget set by, say, some department or manger. They will have to apply for more funding. Or they may have exceeded their overall personal budget and become short of money.
3 It could be a way of expressing the desire for prompt payment, but it depends on how sympathetic you think your debtor is.
Again, this is the opinion of one BrE speaker, others may disagree and it may mean something different to an AmE speaker.