What is the difference between: “I am a shareholder as on Record Date and undertake to hold RE till issue closing date” and " I am renouncee and hold rights entitlement in valid demat account to hold RE till issue closing date".

I am a shareholder as on Record Date and undertake to hold RE till issue closing date

This means that you hold the shares in your demat account as on the record date and you are eligible to get rights entitlement and hold it / sell it / apply for rights shares till issue closing date. Select this option if you got the credit of rights entitlement consequent to you being the shareholder on the record date.

I am renouncee and hold rights entitlement in a valid demat account to hold RE till issue closing date

This means that you have purchased rights entitlement from the secondary market in the cash segment of the Exchange. Hence you bought it from someone who has renounced his rights. The seller would be the renouncer and you would be the renouncee. Select this option if you purchased REs from the secondary market.