Were/are there any other terms for "Christian" which don't mean simply "anointed one"?

No, there is no more useful term. The etymology matters. This is so, even if the so named Christians are oblivious to the underlying etymology.

Paul had this to say in Acts 24:14.

“But this I confess unto thee, that after the way which they call heresy, so worship I the God of my fathers, believing all things which are written in the law and in the prophets".

We see above, the leaders of the people at one time viewed the Disciples as being another way, or sect. Judaism was prone to division and strife, this particular example was just heretical.

We can go back further to Acts ch.4 to see how this fracturing from fringe sect, to separate faith begins.

⁷ And when they had set them in the midst, they asked, By what power, or by what name, have ye done this?

⁸ Then Peter, filled with the Holy Ghost, said unto them, Ye rulers of the people, and elders of Israel,

⁹ If we this day be examined of the good deed done to the impotent man, by what means he is made whole;

¹⁰ Be it known unto you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom ye crucified, whom God raised from the dead, even by him doth this man stand here before you whole.

¹¹ This is the stone which was set at nought of you builders, which is become the head of the corner.

¹² Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved."

The etymology matters here because Peter invokes "Kristos". The men he is addressing are also Kristos, because some are of the Priesthood, anointed to minister. Peter is claiming association, and representation of a separate kristos, Jesus of Nazareth, who is not of the lineage of Levi.

This fracturing does not occur at once, but over time. As in ch.4, the Apostles testify again in ch.5. This also functions as a second indictment of the Priesthood. Again they claim Kristos, a separate Kristos. The leaders understand the meaning of this use of kristos, and the blood guilt on their heads:

²⁷ And when they had brought them, they set them before the council: and the high priest asked them,

²⁸ Saying, Did not we straitly command you that ye should not teach in this name? and, behold, ye have filled Jerusalem with your doctrine, and intend to bring this man's blood upon us.

²⁹ Then Peter and the other apostles answered and said, We ought to obey God rather than men.

³⁰ The God of our fathers raised up Jesus, whom ye slew and hanged on a tree.

³¹ Him hath God exalted with his right hand to be a Prince and a Saviour, for to give repentance to Israel, and forgiveness of sins.

³² And we are his witnesses of these things; and so is also the Holy Ghost, whom God hath given to them that obey him."

On the other hand, we can look at Islam. If we use the old term "Mohammedan", we loose the concept of what Islam is. Gone is submission, or peace with Allah. Instead is a vague notion of a cultural tie to a long gone historical figure. To say, There is no Allah but Allah, and Muhammad is his Prophet is vastly different. It is Peace, it is Submission. It is Islam.

"Christian" doesn't "mean simply ‘anointed one’". And the anointing was not simply "one aspect of their historic ritual practices". It is a reference to one very specific instance and one very specific person.

While the term "Christian" does have "anointed" as its root meaning, it doesn't mean that "anointed" applies to each individual Christian. The connection is much more indirect.

Anointing with oil was used to sanctify high priests:

“And the holy garments of Aaron shall be his sons’ after him, to be anointed in them and to be consecrated in them.
— Exodus 29:29

Anointing was also used to designate a king, as when David was chosen to replace Saul as the King of Israel:

… And the LORD said, “Arise, anoint him; for this is the one!” Then Samuel took the horn of oil and anointed him in the midst of his brothers; and the Spirit of the LORD came upon David from that day forward. …
— 1 Samuel 16:12–13

The Hebrew scriptures are full of prophecies of the day when a new King would arise, overthrow the oppressors, and bring a golden age of peace and prosperity to Israel. This new king was referred to as "the anointed one", which in Hebrew is "messiah" and in Greek is "christos".

The Greek scriptures are a record of that prophesied anointed one, Jesus, who was referred to as the Messiah or the Christ.

… “We have found Him of whom Moses in the law, and also the prophets, wrote—Jesus of Nazareth, the son of Joseph.”
— John 1:45

Jesus's followers became known as Christians, because they believed that Jesus was the prophesied messiah, the christos, anointed as both king and high priest to the world.

This is an example of the Etymological Fallacy (Wikipedia).

Language is an adapting, evolving process wherein the original etymology of a word is often departed from in continuing usage to the point where the actual meaning of the word changes to something other than its original origin : SE-EL&U..

There are further archives searchable in SE-English Language and Usage on this subject.

Yes, 'Christian' comes from a Greek root meaning 'anointed'. But 'Christian'in modern parlance does not mean that anymore. Except when a particular group of persons, say in a congregation, specifically use it among themselves in a way which they, themselves, have decided it shall be used.

The term you are looking for is 'saints' which is used throughout the Authorised Version of the bible (about 160 times, according to Robert Young's 'Analytical Concordance') to translate agios, αγιοσ - Strong 40 which means 'holy ones' or 'separated ones'.

This term was frequently used in history but in modern times it has been dropped, being only used to refer to specific persons regarded as special in some way, whereas in scripture and in history the word was used to denote all believers in Jesus Christ, or (prior to his incarnation) to refer to those who looked for the coming of Messiah, who is, himself, the 'anointed one'.

Historically, those who followed Jesus Christ were seen to be separated from the world and holy unto the Lord. This concept has changed in modern times.

Your question highlights that there is, indeed, a strong connection (in scripture) between 'saints' and 'the anointed'. Which might well lead to a further question on the subject, I would hope.