I do agree that in some respects these countires can be said to be "communist by-name only", however it is important to understand that regardless of that the ruling parties do still have very far-left economic policies. Furthermore, I interpret pure communism as an idealistic end-goal rather than a legitimate economic system - ever since the emergence of communism in the geopolitical landscape, states such as the USSR had more of a socialist economy with an emphasis on state ownership and power. No country has managed to reach the end-state of communism as Marx had envisaged, and really created their own interpretation of Marxism to suit their own bureaucratic needs.

There are only a few states left with a communist governing party as the sole sovereign power - I believe they are China, North Korea, The Republic of Cuba, Laos and Vietnam. Some of these countries are one-party states and have varying degrees of authoritarianism in their governments. Also, some of these states such as China have adopted some changes to their policy and have relaxed some of their communist principles, for instance, some of these countries have recognised private property. Nevertheless, in all of these nations it can be said that the state do have a large degree of control and power.

It is also noteworthy that there are a lot of democracies with communist parties that have a significant degree in legislative power, though these are not considered communist states because other parties also hold (typically more) voting power.

It truly depends on the nation in question. Many states in the first world with a fundamentally free market economy would be more liable to corporate influence, where the legislative would be more keen to draft statutes which protect corporate interests or impose less regulations, and in some nations the laws behind lobbying to protect these interests are less strict. In this context, it could be argued in an abstract sense that corporations have an undue influence on the sovereign, but naturally the term power carries different connotations to it - while they may have the resources and the favour of the government generally, the power to legislate is still in the hands of the sovereign alone despite this influence. It is also worth mentioning that first world countries typically have a system of democracy in which, if corporate influence on the legislative powers becomes too problematic, the party under corporate influence can be voted out of government.

On the flip side, corporate power is far less pronounced in left-leaning countries. Social democracies tend to impose tighter regulations on corporations, and in far-left administrations (especially communist regimes the power of the state is almost absolute.