Competitive vs complete and non-competitive vs incomplete marekts
Complete market is a market where every possible asset or good can be assigned a price and where you have perfect information, can make perfect contracts and zero transaction costs. Any market can be complete regardless of its market structure. So you can have complete market dominated by monopoly, or oligopoly or monopolistic competition etc.

Perfectly competitive market is a market where there are so many buyers and seller that firms become price takers and when the products are identical so there is a little scope for differentiation.

You can’t just call non-competitive market incomplete. For example, in standard undergrad treatment of monopoly you still assume complete markets, yet the market structure is non-competitive.

Yes when people talk about complete markets they do it in a pretty much same sense as Arrow-Debreu