Buying a proxy means letting a stranger sit in the middle of your connection. That is not a reason to avoid it, but it is a reason to know what they see.
What the operator sees
The fact of a connection, the time, the volume and the destination. In other words: which of your machines connected to which Telegram servers, when, and how much data moved. Enough to work out how many accounts you run and on what schedule.
What they do not see
Content. Telegram encrypts the traffic before it reaches the proxy, and the proxy forwards something already sealed. Messages, contacts and files are not readable from proxy logs.
The exception is if somebody talked you into installing their certificate or using a «special» client. Do not: that is precisely when content becomes visible.
What «no logs» is worth
Exactly the seller's reputation. You cannot verify it: logs are either kept or not, and from the outside the two are indistinguishable. Treat the phrase as an intention, not a guarantee.
Practical conclusions
Do not run work accounts and personal ones through the same proxy — the link between them is visible to whoever owns the address. Avoid free and shared proxies entirely: there the business model usually is the traffic. Split client projects across different suppliers.
What else leaves your machine besides traffic is in network fingerprint.
Storing credentials
Proxy logins are keys, exactly like session files. Keep them the same way — the session article applies to them in full.
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