Did Julius Caesar reduce the number of slaves?

No, Julius Caesar probably not reduced the number of slaves.

Slavery was an essential component of ancient civilizations, not just Rome. In Rome, poverty was described as not being able to afford one slave.

Slave owners gave some slaves their freedom, this was called manumission. Those now free men (and women) did not become Roman citizens, but freedman. As such, they were still legally loyal to their former owner. They couldn't marry without his permission, don't do any work he didn't want them to do, etc.

Manumission happened on special occasions, for example at the death of their owner, to celebrate a marriage or coming of age of the eldest son. Becoming dictator could be such an occasion.

Even very large slave owners, such as Caesar, would manumit relatively few slaves. Though in Caesar's case, that would still amount to hundreds, if not a couple of thousand, slaves. Usually slaves in better positions in the household.

Nowhere in the Roman Empire was the abolishment of slavery an issue. Even Spartacus didn't have a problem with it. He forced captured Roman senators to perform gladiatorial fights in the arena. Not exactly what you expect from a man who is perceived to be the enemy of slavery. And he was the only one we remember (incorrectly) in the Ancient world for that.

As the economy was based on slavery, it is extremely unlikely Caesar would manumit many thousands of slaves, just because he could. Caesar had nothing to gain from that and much to lose.

Fellow senators wouldn't be happy, having to follow his example. Ordinary Roman citizens (something Caesar cared a lot about) would be furious. Their status would go down, and they had to compete against those freedmen.

Caesar manumitting thousands of slaves would be very exceptional, something that would be mentioned in most sources. Living in a colony was not something poor Roman citizens looked forward to. Expanding citizenship (which Caesar definitely did) was always a hot potato in Roman politics.

Did Caesar reduce the number of slaves?

Possibly, but we have no convincing evidence or reliable statistics with which to make such a statement with any certainty.

The passage you cited seems to have been at least partly derived from these excerpts of Suetonius' Life of Caesar (42):

...the assignment of eighty thousand citizens to colonies across the sea, ....those who made a business of grazing should have among their herdsmen at least one-third who were men of free birth. He conferred citizen­ship on all who practised medicine at Rome, and on all teachers of the liberal arts, to make them more desirous of living in the city and to induce others to resort to it.

Caesar was concerned that there were too many slaves to be properly supervised in the countryside. Noted historian Richard Billows writes:

… since the great slave uprising under Spartacus it had been clear that the enormous scale of the slave economy, and harsh conditions along with lax supervision, were a serious problem, yet nothing had been done about it. Caesar now ruled that one-third of all herders in the Italian countryside must be freeborn, thereby ensuring much better supervision of the slave herdsmen, improving conditions for them since they would inevitably have similar working and living conditions to their free fellow herders, and provided work for many thousands of impoverished free Italians.

Source: Richard A. Billows, 'Julius Caesar, the Colossus of Rome' (2009).

Although his reforms may have led to a reduction in the number of slaves in herding, freeing them would not make them freeborn so they were presumably assigned elsewhere, unless they were made freedmen. Either way, there is not much (if any) evidence that a significant number of slaves were actually freed. Rather, the aim seems to have been to try to lessen the chances of future slave revolts by closer supervision and better conditions rather than by reducing numbers.

Somewhat ironically, Caesar himself was partly responsible for the large slave population as, during the Gallic wars (58 to 50 BC), he had enslaved huge numbers of Gauls; Plutarch put the number at more than 1 million, though many of these probably did not end up in Italy and Plutarch may well have exaggerated. The whole area of slave numbers in Rome is hugely speculative for

… hardly any genuine statistics are available …

We do not know the number of slaves in any particular community of Roman Italy or in a particular sector of the economy at any given point in time, let alone for the region as a whole.

Source: Walter Scheidel, 'The Roman slave supply' (Princeton/Stanford Working Papers in Classics, May 2007).

On the issue of citizenship, the article you cited is on firmer ground but we should be careful not to exaggerate the extent to which he granted citizenship. According to Dio 41.36.3, in 49 BC, Caesar

… to the Gauls living south of the Alps and beyond the Po he gave citizen­ship because he had once governed them.

Also,

Caesar awarded Roman citizenship to the people of the Spanish town Gades in the same year (Liv. Per. 110; Dio 41.24.1.), but this in itself tells us little about his attitude to any large-scale extension of Roman citizenship beyond Italy. Brunt goes only so far as to say (1971: 239), “It is agreed that Caesar was much more ready than Republican statesmen had been to enfranchise provincials,” and he cites the Trans-padani, Gades, and “conceivably some other provincial towns.”

Source: Miriam Griffin (ed), 'A Companion to Julius Caesar' (2009).

Elsewhere,

In the East, no Greek cities received either citizenship or Latin rights, though several prominent Greeks and others received individual grants (Cic. Fam. 13.36, Phil. 13.33; Plut. Cic. 24). So did teachers of the liberal arts and doctors, many of whom will have been Greeks, settling in Rome (Suet. Iul. 42.1).

Source: Griffin