For Feb 1, I would say ‘bring forward’, not push forward (you push something away from you, not towards you).
For Feb 5, postpone or push back are both OK in terms of meaning; postpone is more formal so it would be better suited for a business meeting rather than a dinner with your friend.
Postpone sounds ok. Push back/push forward not so much. I’m sure you would be understood but Postpone sounds better. Can we postpone our dinner until Feb 5th? For bringing it forward, I would simply ask "Can we reschedule? Or Can we reschedule for an earlier date…how about Feb 1st? “Reschedule” could be used in both situations.
When arranging meetings with friends or colleagues you really don’t want any misunderstandings. Best of all would be use frequently used terms such as reschedule or postpone as already mentioned such as in the example “Can we postpone our dinner until Feb 5th?” -why bother with less used terms which might cause confusion?
In British English we can say, for example “let’s meet at half 4” (which means at 4.30 pm). In European countries such as The Netherlands, Denmark or Germany “half 4” would be “half 5” - which causes confusion all round. You may find people remembering there is an hour’s difference in the two languages and turning up at 3.30 getting the hour the wrong way around, whilst others could turn up at 5:30. Certain European speakers do not think that half 4 means half past four means 4:30. I prefer to avoid confusion whenever possible and never use half 4 with foreigners as I know they’ll think “half five” in their own language (“halv fem”, half fünf" or 4.30 pm) whereas half five said by a Brit means 5.30 pm - likewise I prefer to “postpone” or “reschedule” a meeting - it’s less complicated that way!