Both concepts refer to different levels of government. For the U.S., for example, central government debt refers to the debt by the federal government, while general government debt refers to overall debt, including states and municipalities. Depending on how fiscal federalism is designed, different levels might be affected differently by the business cycle.
The OECD offers the same indicators, with some information on how they are defined. I copy from there:
Data are within the framework of the System of National Accounts (SNA), a set of internationally agreed concepts, definitions, classifications and rules for national accounting. Using SNA terminology, general government consists of central, state and local governments, and social security funds. State government is only applicable to the nine OECD countries that are federal states: Australia, Austria, Belgium, Canada, Germany, Mexico, Spain (considered a quasi-federal country), Switzerland and the United States. Debt is a commonly used concept, defined as a specific subset of liabilities identified according to the types of financial instruments included or excluded. Generally, it is defined as all liabilities that require payment or payments of interest or principal by the debtor to the creditor at a date or dates in the future. All debt instruments are liabilities, but some liabilities such as shares, equity and financial derivatives are not debt.
Debt is thus obtained as the sum of the following liability categories, whenever available/applicable in the financial balance sheet of the general government sector: currency and deposits; debt securities; loans; and other liabilities (i.e. insurance, pension and standardised guarantee schemes, other accounts payable, as well as, in some cases special drawing rights - SDRs). According to the SNA, most debt instruments are valued at market prices, when appropriate (although some countries might not apply this valuation, in particular for debt securities).
The treatment of government liabilities in respect of their employee pension plans is diverse across countries, making international comparability difficult. In particular, according to the former 1993 SNA, only the funded component of the government employee pension plans was reflected in its liabilities. However, the 2008 SNA recognises the importance of the liabilities of employers’ pension schemes, regardless of whether they are funded or unfunded. For pensions provided by government to their employees, countries have some flexibility in the recording of the unfunded liabilities in the core tables; this has also been followed by the ESA 2010, its European equivalent (although a new supplementary table will be added showing liabilities and associated flows of all pension schemes, whether funded or unfunded). Some OECD countries, such as Australia, Canada, Iceland, Sweden and the United States (including others whose data source is the IMF Economic Outlook) record employment-related pension liabilities, funded or unfunded, in government debt data. For those countries (except non-OECD ones), an adjusted government debt ratio is calculated by excluding from the debt these unfunded pension liabilities. Government debt used here is recorded on a gross basis, not adjusted by the value of government-held assets.
The SNA definition of debt differs from the definition applied under the Maastricht Treaty, which is used to assess EU fiscal positions.
Gross domestic product (GDP) is the standard measure of the value of goods and services produced by a country during a period.
The reason for countries reporting only one indicator could be that the respective statistical office simply only report one number. For small countries, general government debt might be by and large equated with central government debt if there is no relevant fiscal capacity below the central government.
The explanation for Cyprus might be that the central government had to bail out banks recently, leaving the non-negligible budgets of municipalities unaffected, but this is a bit speculative.