Do you have space to store an extendable table when it's closed, or would you prefer something that's always on display?
Because honestly, extendable tables are the most versatile option if you're planning for the dining room to have two lives: one compact everyday version and another for when friends come over. The thing is, you need a corner where you can tuck the extension when you're not using it, or generous space that can handle the table open without it looking like you live on it.
Round tables have their charm in small spaces because visually they "breathe" more - they don't have those corners that make a rectangular one feel invasive - but they basically take up the same area as an equivalent square table. What they do gain is functionality: nobody gets stuck at the head, people circulate better around it, and you can fit one more chair without it looking ridiculous.
If yours is a really tight apartment and you normally eat alone or with one other person, maybe a small fixed rectangular table is the most sensible option. It takes up little space, doesn't need maintenance, and if you ever need extra space for something occasional, you can always pull out a tray or a side table. Extendable tables sound good in theory, but they require discipline so you don't end up with the extension stored on your living room armchair.