Is there a general understanding of worship and a particular understanding?
Yes, the general understanding of worship is what I indicated previously. It is having humility to Allah out of love and magnification by doing what He commands and avoiding His prohibitions, in a manner the religion has legislated. This is the general meaning (of worship).
The particular understanding - meaning with its details - is what shaykh al-Islām ibn Taymiyah mentioned: Worship is a comprehensive term including everything that Allah loves and is pleased with from statements and actions, inwardly and outwardly. Just as fear, awe, reliance, prayer, paying zakat, fasting and other than this from the rulings of Islam.
Also if you intend (from your question) the particular and general understanding that some of the people of knowledge mentioned, that worship is either universal or legislative which means that man perhaps he is submissive to Allah universally or legislatively, then the universal submission is general. This includes the believer and the non-believer as well as the goodoer and evildoer. As mentioned in the statement of Allah:
إِن كُلُّ مَن فِي السَّمَاوَاتِ وَالْأَرْضِ إِلَّا آتِي الرَّحْمَٰنِ عَبْدًا
“There is no one in the heavens and earth but that he comes to the Most Merciful as a servant.” (Maryam: 93)
Therefore every thing in the heavens and on the earth is submissive to Allah under His universal will and its not possible for anyone to oppose or contradict Him in what He wants, with His universal decision.
As for the particular worship, then it is the legislative worship. It is submission to Allah legislatively. This is particularly to the believer in Allah, carrying out His command. Then there is more particular above that. So the more particular type of worship is that of the Messengers. As in the statement of Allah:
تَبَارَكَ الَّذِي نَزَّلَ الْفُرْقَانَ عَلَىٰ عَبْدِهِ
“Blessed is He who sent down the Criterion upon His Servant...” (al-Furqān:1)
And His statement:
وَإِن كُنتُمْ فِي رَيْبٍ مِّمَّا نَزَّلْنَا عَلَىٰ عَبْدِنَا
“And if you are in doubt about what We have sent down upon Our Servant” (al-Baqarah:23)
And His statement:
وَاذْكُرْ عِبَادَنَا إِبْرَاهِيمَ وَإِسْحَاقَ وَيَعْقُوبَ
“And remember Our servants, Abraham, Isaac and Jacob...” (Sad: 45)
and other than these (verses) which describe the messengers with servitude (worship).
Source: Fiqh al-Ibadaat pg.13
Translated By Abu Anas Atif Hasan