In the context of the evolving energy mix of power systems, marked by the growing integration of renewable energy sources and battery energy storage systems (BESS), the dynamics of power systems are undergoing significant changes. In such systems, dynamic couplings between inverter-based resources (IBRs) and synchronous machines (SM) may occur, leading to stability issues. Stability concerns may also emerge due to communication delays between external control loops and inverter's reference input, interacting with both the inverter's primary frequency control and the frequency meter (FM) used to measure the grid's electrical frequency. The current paper analyses and studies these phenomena on a use-case based on a French island power system. This system may experience oscillations among its generation sources, causing concerns for the grid operator. Therefore, there is a need for comprehensive studies to highlight the phenomena that could compromise stability and operational safety, especially considering the significant number of new renewable energy and BESS projects that are currently under development. For this purpose, an inherited simple and sufficient conceptual benchmark is used with some adaptations to model the island power system. This benchmark, composed of an inverter, a synchronous machine, and an equivalent dynamic grid model, has been linearized around various operating points, and modal analysis applied to detect and analyse all the oscillation modes. Also, participation factors are used to gain insights on the type of each mode. Through modal analysis, sensitivity studies are conducted to assess the impact of key parameters on the damping of newly identified oscillation modes. The linearized benchmark is validated through nonlinear time-domain EMT-based (electromagnetic transient) simulations using commercial EMT simulation tools. Two new modes of oscillation are detected in addition to the traditional inter-area mode and average frequency mode. The first one is caused by interactions between the synchronous machine and the inverter and is called coupling mode. The second one, called control delay mode, is mainly due to an interaction between the communication delay and the inverter's primary frequency control. Finally, a Power System Stabilizer (PSS) has been tuned and installed on the SM to damp the inter-area mode, and a Power Oscillation Damping (POD) controller added to the inverter's power control loop has been proposed and proven to be effective in damping the average frequency mode.