Your May rent likely is late if you have not paid by 2 May. It may be that the landlord would not charge late fees until the 5th of the month. However, your payment is due on the 1st of the month. If you have not paid by the 2nd, you are late.
If you do not pay all the rent that is due as indicated on the notice, the landlord will proceed with the next step of the unlawful detainer proceeding.
Unless you have been a very good tenant (who is running into a little financial rough spot), the landlord likely wants you out of the rental to avoid any more headaches.
The 3 days to pay or vacate are calendar days. So, on 6 May, the landlord will proceed to the next step if not fully paid.
You should review the specific facts with your attorney to find out your legal options.
Almost all leases require rent to be paid by the first, but do not impose a late fee until later in the month, often the fifth. I recommend you review your lease to be sure rent is not late on the second.
The landlord was wrong to refuse your April rent on the 29th, but that does not excuse your duty to pay. You should at least tender the April rent again before the notice period is over. If May rent was due, you should tender both April and May.
A three day notice means three business days. If the notice was hand-delivered to you on May 2, you have until May 5 to pay, the day you said you will have May's rent. If the notice was delivered any other way, you get one additional day until May 6. Be sure to make a record of when and how much you offer to pay so you can prove what they refused to take and when if the case goes to court.