Explanation
No. A Wi-Fi connection and an internet connection are different things. Your device can join a wireless network even when it cannot reach the internet.
TP-Link explains that connecting to Wi-Fi establishes a link with the router, without guaranteeing internet access. Apple's support documentation also explicitly addresses devices that are connected to Wi-Fi but cannot get online.
The problem may involve the internet provider, network equipment or an individual device. The Wi-Fi indicator alone cannot identify the cause, and moving closer to the router does not establish that the wider internet connection is working.
Key takeaway
Connected to Wi-Fi does not necessarily mean connected to the internet. The local wireless link is only one part of the connection.