Does titanium rust?

The word "rust" is reserved to the ferrous metals, which Ti is not.

Ti does however oxidize in air the same way as Al and Mg. The resulting TiO2 forms a coating that prevents more oxidation.

Ti oxidizes but does not rust .
Rust is when both moisture and oxygen react with only iron
The rusting of iron is an electrochemical process that begins with the transfer of electrons from iron to oxygen.[4] The iron is the reducing agent (gives up electrons) while the oxygen is the oxidising agent (gains electrons). The rate of corrosion is affected by water and accelerated by electrolytes, as illustrated by the effects of road salt on the corrosion of automobiles. The key reaction is the reduction of oxygen:

O2 + 4  e− + 2  H2O → 4  OH−
Because it forms hydroxide ions, this process is strongly affected by the presence of acid. Indeed, the corrosion of most metals by oxygen is accelerated at low pH. Providing the electrons for the above reaction is the oxidation of iron that may be described as follows:

Fe → Fe2+ + 2  e−
The following redox reaction also occurs in the presence of water and is crucial to the formation of rust:

4 Fe2+ + O2 → 4 Fe3+ + 2 O2−
In addition, the following multistep acid–base reactions affect the course of rust formation:

Fe2+ + 2  H2O ⇌ Fe(OH)2 + 2  H+
Fe3+ + 3  H2O ⇌ Fe(OH)3 + 3  H+
as do the following dehydration equilibria:

Fe(OH)2 ⇌ FeO +  H2O
Fe(OH)3 ⇌ FeO(OH) +  H2O
2 FeO(OH) ⇌ Fe2O3 +  H2O

Iron oxide is not rust

No. Only ferrous metals rust.

No only iron rusts

It can happen but the metal is more resistant to oxidation than iron

No.

No , Ti does not rust like Fe since it makes a very thin passive layer of its oxide which prevents its corrosion or rusting

Hell yes!!! Got some in my knee replacement. Now when I walk it squeeks!! Spray my knee with WD40, but it don't hep a bit.

no, only iron or iron containing metals rusts, making iron oxid

Nope, Ti is virtually indestructible.