Hi Rist,
I am doing the same arduino+ESP8266 (AT way) until now 451000+ points in TS - it works and for less than $10
1 tried via browser (you too)
2 tried directly with ESP8266 via RS232 with Termite (same as arduino monitor) typing AT commands and waiting answers then check the plot in my TS channel
3 writing the arduino program to plot fully automatic
What I did for point 2
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AT+GMR
AT version:0.50.0.0(Sep 18 2015 20:55:38)
SDK version:1.4.0
compile time:Sep 18 2015 21:32:07
OK
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AT+CWMODE?
+CWMODE:3
OK
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AT+CIPMODE?
+CIPMODE:0
OK
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AT+CIPMUX?
+CIPMUX:0
OK
------------------------------------------
AT+CIPSTATUS
STATUS:2
OK
------------------------------------------
AT+CIPSTART="TCP","api.thingspeak.com",80
CONNECT
OK
------------------------------------------
AT+CIPSEND=64
OK
>
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GET /update?api_key=sixteenUPlowcase&field1=25.12&field2=55.31 << count 1 space after GET before slash totaling 62 plus [newline][return]=64
Recv 64 bytes
SEND OK
+IPD,5:12825CLOSED << 3 lines TS say it was fine
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Note when I did
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AT+CIPSTART="TCP","184.105.153.149",80
ERROR
CLOSED
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it DID NOT worked!! (I do not know why ... IP?)
Count spaces and follow UPPER/lower case ... (use non proportional fonts I use Notepad with CONSOLAS font to SEE spaces)
I did this a few times for 2 days ... just in casw
I did not checked carefully your program but in loop you are using 2 seconds not the 15 minimum I use 1 minute (Good to me)
I tried at least 2 versions of firmware and syntaxis varied slightly. i.e. AT+CIPSTART answer ERROR and Error so arduino gets confused
Can you post your serial output?