Making the case that Pro-Liberty Conservatives rightfully deserve to be called Liberals.
A proper discussion should entail two primary points of contention.

• Classic Liberals are essentially Liberals.
• Conservatives are in essence classic liberals.

Let’s begin with looking at how the various dictionaries handle the phrase ‘classical liberal’ compared to the original term. In most cases, online dictionaries will default to the original term when searching for the definition of the phrase. Thus, according to these reference sources, ‘classical liberals’ are merely Liberals QED. This should make the case in general terms that Classic Liberals and Liberals are one in the same.
The word ‘classic’ also tries to make reference to the original meaning of the word, but that really isn’t the case with most definitions. In many ways, that phrase distinguishes ‘classic Liberals’ from ‘modern Liberals’, but since the term is closely associated with Liberty, just the word Liberal should make the case.

The false argument that ‘Word meanings have changed’

There are those who try to use the ‘word meanings have changed’ excuse to justify hiding behind that laudatory moniker. The problem for them is that really isn’t the case. In point of fact, most words haven’t changed, despite the post modernism view on the subject. A simple test will help prove the point, take a dictionary and open it to a random page. The odds are very good that most of not almost all the words on the page will have not undergone any change over the past few years.
One must also consider that the terms Liberal and Liberty are very closely related in sound and meaning. Both have the same roots in the term Libertas, the female personification of liberty and personal freedom.
The term Liberty would also have to change along with word Liberal. Liberty still has the same meaning, therefore the term Liberal remains unchanged.

Leftist exploitation of the term Liberal while being hostile to Liberty

Many on the Left have the infuriating habit of hiding behind various deceptive terms, with their self-labeling as Liberal being no exception. They persist in using this term for themselves while displaying open hostility towards the cause of Liberty. In many ways it’s the height of deceit to take on this label while working against individual freedom.
It stands to reason that those using a term grounded in ‘liberty and personal freedom’ should be proponents of those concepts and not stand in opposition to them. As some have observed, Leftists prefer to have that laudatory moniker attached to them because they can carry the mantle of being Liberators.

One cannot be a Liberal while acting in opposition to Liberty

The word infuriating in describing the Left’s exploitation of because in a process that seems to be accelerating by the day, the Left is increasingly going after the cause of Liberty. One only need a short perusal of the day’s news to see these assaults, ranging from attacks on free-speech in the guise of the arbitrary term ‘hate speech’, to outright censorship with ‘de-platforming’. Then there are incessant attacks against the common sense human right of self-defense that are coming in from every direction. Not to mention attempts at controlling personal property or their work at depriving people of basic privacy.

Contrast Leftist attacks against Liberty with those on the Right trying to conserve it

Despite the falsehoods foisted by the Left, Conservatives are the true champions of Liberty. We are on the side of defending the cause while the Left is working day and night in coming up with new ways of restricting freedom.
This used to just be in the field of self-defense, with the Left constantly assaulting this basic human right. Even that has changed on an accelerating basis with their branching out with new rules on ‘hate speech’ or insane levels of language control with the ever-changing agenda of political correctness.
What has become even more disturbing is that Leftist assaults on Liberty entail multiple parts of the Bill of Rights. The newest outrage are so-called ‘red flag’ laws, with freedom of speech, self-defense and due process on the chopping block. Not to be outdone, there are some Leftists who have proposed tying a Constitutionally affirmed right to the idea of ‘thoughtcrime’, demanding searches through one’s social media accounts and Internet inquiries.
Most Conservatives are ‘resisting’ (to coin a phrase) this level of Leftist tyranny. We are the ones on the side of Liberty, thus we are the ones deserving of the term Liberal.

At least one Leftist admits that Classic Liberalism is Conservatism

Quite often it’s useful to listen to those hostile to Liberty on the Left. YouTube in it’s wisdom usually has recommendations for follow on videos. In the case of those on ‘Classic Liberalism’ there is one entitled “Classical Liberalism” is Pretty F&^#@%g Conservative!” a ‘long form deep dive into classical liberalism’.  Making the case that Classic Liberalism is in essence Conservatism, minus the stand in expletives.

Consider a thought experiment on the alleged changing meaning of the word Liberal.

Let us ponder on the question of Leftists being Liberal by the supposedly malleable meaning of the word. If they are allegedly of that stripe, given their recent agenda items and demands, it would seem that a Liberal is one against freedom of religion, freedom of speech, freedom of the press or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.
Add to this that said ‘Liberal’ in the world of new-speak would be against the Human Right of self-defense, the presumption of innocence and due process. We could go on, but the point should be clear.
Does this sound like anyone championing the cause of Liberty? If that isn’t the case, then those people aren’t Liberal. Those conserving Liberty are the true Liberals.
It is a matter of course that Leftists will wholeheartedly support these natural rights for those they favor, but that isn’t the point. Those rights are also for those they oppose, when have they done that as of late? When have they spoken of the aphorism that they may ‘disagree with what some has to say, but they will defend to the death their right to say it?’

One final note from Ronald Reagan on the close connection of Conservatism, Liberalism and Liberty

President Ronald Reagan tangentially made this case in a portion of his farewell address 1/11/89:

“I hope we have once again reminded people that man is not free unless government is limited. There’s a clear cause and effect here that is as neat and predictable as a law of physics: As government expands, liberty contracts.” [emphasis mine]

I re-emphasize, “as government expands, liberty contracts.”
Consider in general terms which side of the political spectrum that wants to expand government while denigrating Liberty. Then consider which side ‘resists’ this expansion while defending Liberty. Most people should intuitively know that ever expanding government control always has a cost in freedom.
Now, which side has the better claim on the defense of Liberty and the praiseworthy moniker of Liberal? Is it the Left with an ever accelerating zeal to expand government or the right trying to limit it’s reach and conserve Liberty?