Caesar was originally a family name, adopted by Augustus and his Julio-Claudian successors in memory of Augustus' adoptive father Gaius Julius Caesar. For Augustus it established his legitimate familial descendancy from Julius Caesar, and for Augustus' successors it established their legitimate political descendancy from Augustus himself. It took on a different form when the Julio-Claudian dynasty ended in AD 68, and Vespasian adopted it and after this point it would have to be a considered a piece of standard imperial nomenclature, along with other terms such as Imperator or Augustus.

The meaning of the term changed again to mean the heir or second in command to the Emperor, solidifying under the reforms of Diocletian, where it became a technical term for the junior partner to the Emperors in the East and West empires: the senior emperor was the 'Augustus' and the junior was the 'Caesar'. This system, termed the 'Tetrarchy', lasted until the early 300's AD. However, during this entire time the name/word 'Caesar' had also remained part of the nomenclature for all Emperors. It only seems to have dropped out of useage from the time of the Valentinian dynasty (AD 364-392) to the end of the empire, though I'm not sure why. 'Augustus' seems to have replaced it totally by this stage, and always appears to have been the more important term.

Essentially, it was never specifically a military title, but those with the name were effectively either commander-in-chief of the Roman military or the most senior military field commander in the Empire. It was probably closest to a military title under the Tetrarchy, when a key point of each Augustus having a junior Caesar was to have somebody trustworthy who could carry out major campaigns on their behalf (Austin & Rankov 246). After this point, as far as I know the magister militum seems to have filled a similar role. Interestingly, it was under Diocletian that the separation of military and civil posts was made. Provincial governors retained their title, but they now had a dux under them who controlled the military of the province. Later Emperors did not make major changes to this civilian/military split of command in the provinces (Southern 189). Perhaps an echo of this provincial system can be seen in his appointment of junior Caesares to the two Augusti. Therefore, if there was any period wherein 'Caesar' was a military title, it was at under the Tetrarchy, but it really depended on the Augustus-and-Caesar-team in particular how they split military control.

I'm not sure "commander in chief" is a specifically military title, rather a dual civilian-military title emphasising supreme civilian control over the military. As the management of tension between the Emperor and the Army (or rather key generals) was key to the stability of the Empire at any given time, you might say the the US title of "commander in chief" as held by the President is everything a Roman Emperor could have wished for in terms of control over the Army. However, the civilian government and the military was not nearly so cleanly separated in the Roman Empire as it is today in the US, and so while the US President can control the military with what is basically a civilian title and position, the Roman Emperor needed to rely upon direct personal loyalty from the army (and especially close control over its commanders) and more specifically military nomenclature such as 'Imperator' to establish his claim as commander in chief. Therefore, while the term 'Caesar' could be argued to be an effectively military title in the Tetrarchy, the Caesares were always junior to the Augusti and so it was never equivalent to the US President's position as "commander-in-chief".

Sources:

NJE Austin and NB Rankov, Exploratio: Military and political intelligence in the Roman world from the Second Punic War to the battle of Adrianople, London: Routledge 1995.

Patricia Southern, Ancient Rome: The Empire 30 BC- AD476, Stroud: Amberly Publishing, 2009.

No. Caesar was a name that, due to Julius Caesar and Augustus' hold over the Army, had enormous clout with Roman soldiers. It also became synonymous with the post of Princeps (emperor) over the course of the Julio-Claudian dynasty to the point where all the non-Julio-Claudian emperors took it as a mark of emperorhood.

The term you're looking for is imperator: one who has imperium. By the period of the principate, this had come to encompass executive power over a portion of the state, but its technical meaning was the right to command soldiers. Many Roman magistrates could hold imperium, including Praetors, Proconsuls, and Consuls. Additionally to his role as proconsul of aboult half the provinces in the Empire, the princeps held imperium maius - greater imperium - which allowed him to override the imperium of any other magistrate, but this right was only exercised when absolutely necessary, since a key point of the principate was that, while it was an autocracy, it should't look like one. That's the closest thing to a 'commander-in-chief' title that the Roman empire had, although the legal power was backed up by the fact that the Emperor was mostly in charge of paying the soldiers, and thus commanded their loyalty. (Nero's failure to pay and care for the Army was a key factor in his loss of control of the Empire)