As with many things, the answer is complicated. Some initial thoughts: Both parties struggled to address the concentration of wealth and power that had been happening.

"Progressivism" is a term that is applied to people with some significant differences but for this discussion, I'll apply it to those who wanted government to take action to address the concentration of corporate power and wealth.

There were elements of both parties who came to the conclusion that government should take action. This was more true on the Republican side than Democratic, as to this point Republicans had been the party of using government power for public benefit (Homestead Act, Land Grant Colleges, etc). However, the bulk of those with power in both parties tended to be more classically liberal, as you referenced.

Roosevelt supported breaking up "trusts" if they were running contrary to public interest or abusive to their workers and communities. He didn't think that they all were deserving of being broken up. He preferred strong federal regulation to rein them in.

Wilson campaigned on a platform to break the trusts up. If they were "too big," they were too powerful and should be broken up. Smaller entities would provde more competetion and better service, so "classically liberal" result, but powerful government to get there. Once elected, however, his approach was closer to Roosevelt's plan than his campaign promises.

Taft actually initiated more anti-trust cases than Roosevelt, so kind of acted in a way that was close to Wilson's goals (going after what Roosevelt considered to be "good trusts" was a factor in his 3rd party bid. However, with Wilson and Roosevelt staking out progressive positions and wishing to solidify his partisan base that relied on large corporate support, Taft's campaign in 1912 tended to be more conservative or "classically liberal."

In summary, Roosevelt was Progressive in wanting to use federal power to limit corporate concentration, which carried on some parts of the Republican philosophy. Wilson campaigned on a short-term expansion of federal intervention, which was Progressive, whith the goal of limiting central government power in the long run, as Democrats had preferred to this point. Taft did act as a Progressive in the instance of many anti-trust cases, but was philosophically closer to party leaders who wanted to maintain strong ties to businesses. After World War I, Republicans fully reverted to a non-interventionist (although very cosy and at times corrupt) relationship to businesses. By the time Coolidge came to power, he sought to minimize the intervention and coziness with business.

I hope this helps.